(x+i)(x-i)(x+(1+2i))(x+(1-2i))=x^4+2x^3+6x^2+2x+5

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Solution for (x+i)(x-i)(x+(1+2i))(x+(1-2i))=x^4+2x^3+6x^2+2x+5 equation: 


Simplifying
(x + i)(x + -1i)(x + (1 + 2i))(x + (1 + -2i)) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
(i + x)(x + -1i)(x + (1 + 2i))(x + (1 + -2i)) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
(i + x)(-1i + x)(x + (1 + 2i))(x + (1 + -2i)) = x4 + 2x3 + 6x2 + 2x + 5

Remove parenthesis around (1 + 2i)
(i + x)(-1i + x)(x + 1 + 2i)(x + (1 + -2i)) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
(i + x)(-1i + x)(1 + 2i + x)(x + (1 + -2i)) = x4 + 2x3 + 6x2 + 2x + 5

Remove parenthesis around (1 + -2i)
(i + x)(-1i + x)(1 + 2i + x)(x + 1 + -2i) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
(i + x)(-1i + x)(1 + 2i + x)(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5

Multiply (i + x) * (-1i + x)
(i(-1i + x) + x(-1i + x))(1 + 2i + x)(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5
((-1i * i + x * i) + x(-1i + x))(1 + 2i + x)(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
((ix + -1i2) + x(-1i + x))(1 + 2i + x)(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5
((ix + -1i2) + x(-1i + x))(1 + 2i + x)(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5
(ix + -1i2 + (-1i * x + x * x))(1 + 2i + x)(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5
(ix + -1i2 + (-1ix + x2))(1 + 2i + x)(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
(ix + -1ix + -1i2 + x2)(1 + 2i + x)(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5

Combine like terms: ix + -1ix = 0
(0 + -1i2 + x2)(1 + 2i + x)(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5
(-1i2 + x2)(1 + 2i + x)(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5

Multiply (-1i2 + x2) * (1 + 2i + x)
(-1i2 * (1 + 2i + x) + x2(1 + 2i + x))(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5
((1 * -1i2 + 2i * -1i2 + x * -1i2) + x2(1 + 2i + x))(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
((-1i2 + -1i2x + -2i3) + x2(1 + 2i + x))(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5
((-1i2 + -1i2x + -2i3) + x2(1 + 2i + x))(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5
(-1i2 + -1i2x + -2i3 + (1 * x2 + 2i * x2 + x * x2))(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
(-1i2 + -1i2x + -2i3 + (2ix2 + 1x2 + x3))(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5
(-1i2 + -1i2x + -2i3 + (2ix2 + 1x2 + x3))(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
(2ix2 + -1i2 + -1i2x + -2i3 + 1x2 + x3)(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5
(2ix2 + -1i2 + -1i2x + -2i3 + 1x2 + x3)(1 + -2i + x) = x4 + 2x3 + 6x2 + 2x + 5

Multiply (2ix2 + -1i2 + -1i2x + -2i3 + 1x2 + x3) * (1 + -2i + x)
(2ix2 * (1 + -2i + x) + -1i2 * (1 + -2i + x) + -1i2x * (1 + -2i + x) + -2i3 * (1 + -2i + x) + 1x2 * (1 + -2i + x) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5
((1 * 2ix2 + -2i * 2ix2 + x * 2ix2) + -1i2 * (1 + -2i + x) + -1i2x * (1 + -2i + x) + -2i3 * (1 + -2i + x) + 1x2 * (1 + -2i + x) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
((2ix2 + 2ix3 + -4i2x2) + -1i2 * (1 + -2i + x) + -1i2x * (1 + -2i + x) + -2i3 * (1 + -2i + x) + 1x2 * (1 + -2i + x) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5
((2ix2 + 2ix3 + -4i2x2) + -1i2 * (1 + -2i + x) + -1i2x * (1 + -2i + x) + -2i3 * (1 + -2i + x) + 1x2 * (1 + -2i + x) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5
(2ix2 + 2ix3 + -4i2x2 + (1 * -1i2 + -2i * -1i2 + x * -1i2) + -1i2x * (1 + -2i + x) + -2i3 * (1 + -2i + x) + 1x2 * (1 + -2i + x) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
(2ix2 + 2ix3 + -4i2x2 + (-1i2 + -1i2x + 2i3) + -1i2x * (1 + -2i + x) + -2i3 * (1 + -2i + x) + 1x2 * (1 + -2i + x) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5
(2ix2 + 2ix3 + -4i2x2 + (-1i2 + -1i2x + 2i3) + -1i2x * (1 + -2i + x) + -2i3 * (1 + -2i + x) + 1x2 * (1 + -2i + x) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5
(2ix2 + 2ix3 + -4i2x2 + -1i2 + -1i2x + 2i3 + (1 * -1i2x + -2i * -1i2x + x * -1i2x) + -2i3 * (1 + -2i + x) + 1x2 * (1 + -2i + x) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
(2ix2 + 2ix3 + -4i2x2 + -1i2 + -1i2x + 2i3 + (-1i2x + -1i2x2 + 2i3x) + -2i3 * (1 + -2i + x) + 1x2 * (1 + -2i + x) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5
(2ix2 + 2ix3 + -4i2x2 + -1i2 + -1i2x + 2i3 + (-1i2x + -1i2x2 + 2i3x) + -2i3 * (1 + -2i + x) + 1x2 * (1 + -2i + x) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5
(2ix2 + 2ix3 + -4i2x2 + -1i2 + -1i2x + 2i3 + -1i2x + -1i2x2 + 2i3x + (1 * -2i3 + -2i * -2i3 + x * -2i3) + 1x2 * (1 + -2i + x) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
(2ix2 + 2ix3 + -4i2x2 + -1i2 + -1i2x + 2i3 + -1i2x + -1i2x2 + 2i3x + (-2i3 + -2i3x + 4i4) + 1x2 * (1 + -2i + x) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5
(2ix2 + 2ix3 + -4i2x2 + -1i2 + -1i2x + 2i3 + -1i2x + -1i2x2 + 2i3x + (-2i3 + -2i3x + 4i4) + 1x2 * (1 + -2i + x) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5
(2ix2 + 2ix3 + -4i2x2 + -1i2 + -1i2x + 2i3 + -1i2x + -1i2x2 + 2i3x + -2i3 + -2i3x + 4i4 + (1 * 1x2 + -2i * 1x2 + x * 1x2) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
(2ix2 + 2ix3 + -4i2x2 + -1i2 + -1i2x + 2i3 + -1i2x + -1i2x2 + 2i3x + -2i3 + -2i3x + 4i4 + (-2ix2 + 1x2 + 1x3) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5
(2ix2 + 2ix3 + -4i2x2 + -1i2 + -1i2x + 2i3 + -1i2x + -1i2x2 + 2i3x + -2i3 + -2i3x + 4i4 + (-2ix2 + 1x2 + 1x3) + x3(1 + -2i + x)) = x4 + 2x3 + 6x2 + 2x + 5
(2ix2 + 2ix3 + -4i2x2 + -1i2 + -1i2x + 2i3 + -1i2x + -1i2x2 + 2i3x + -2i3 + -2i3x + 4i4 + -2ix2 + 1x2 + 1x3 + (1 * x3 + -2i * x3 + x * x3)) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
(2ix2 + 2ix3 + -4i2x2 + -1i2 + -1i2x + 2i3 + -1i2x + -1i2x2 + 2i3x + -2i3 + -2i3x + 4i4 + -2ix2 + 1x2 + 1x3 + (-2ix3 + 1x3 + x4)) = x4 + 2x3 + 6x2 + 2x + 5
(2ix2 + 2ix3 + -4i2x2 + -1i2 + -1i2x + 2i3 + -1i2x + -1i2x2 + 2i3x + -2i3 + -2i3x + 4i4 + -2ix2 + 1x2 + 1x3 + (-2ix3 + 1x3 + x4)) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
(2ix2 + -2ix2 + 2ix3 + -2ix3 + -1i2 + -1i2x + -1i2x + -4i2x2 + -1i2x2 + 2i3 + -2i3 + 2i3x + -2i3x + 4i4 + 1x2 + 1x3 + 1x3 + x4) = x4 + 2x3 + 6x2 + 2x + 5

Combine like terms: 2ix2 + -2ix2 = 0
(0 + 2ix3 + -2ix3 + -1i2 + -1i2x + -1i2x + -4i2x2 + -1i2x2 + 2i3 + -2i3 + 2i3x + -2i3x + 4i4 + 1x2 + 1x3 + 1x3 + x4) = x4 + 2x3 + 6x2 + 2x + 5
(2ix3 + -2ix3 + -1i2 + -1i2x + -1i2x + -4i2x2 + -1i2x2 + 2i3 + -2i3 + 2i3x + -2i3x + 4i4 + 1x2 + 1x3 + 1x3 + x4) = x4 + 2x3 + 6x2 + 2x + 5

Combine like terms: 2ix3 + -2ix3 = 0
(0 + -1i2 + -1i2x + -1i2x + -4i2x2 + -1i2x2 + 2i3 + -2i3 + 2i3x + -2i3x + 4i4 + 1x2 + 1x3 + 1x3 + x4) = x4 + 2x3 + 6x2 + 2x + 5
(-1i2 + -1i2x + -1i2x + -4i2x2 + -1i2x2 + 2i3 + -2i3 + 2i3x + -2i3x + 4i4 + 1x2 + 1x3 + 1x3 + x4) = x4 + 2x3 + 6x2 + 2x + 5

Combine like terms: -1i2x + -1i2x = -2i2x
(-1i2 + -2i2x + -4i2x2 + -1i2x2 + 2i3 + -2i3 + 2i3x + -2i3x + 4i4 + 1x2 + 1x3 + 1x3 + x4) = x4 + 2x3 + 6x2 + 2x + 5

Combine like terms: -4i2x2 + -1i2x2 = -5i2x2
(-1i2 + -2i2x + -5i2x2 + 2i3 + -2i3 + 2i3x + -2i3x + 4i4 + 1x2 + 1x3 + 1x3 + x4) = x4 + 2x3 + 6x2 + 2x + 5

Combine like terms: 2i3 + -2i3 = 0
(-1i2 + -2i2x + -5i2x2 + 0 + 2i3x + -2i3x + 4i4 + 1x2 + 1x3 + 1x3 + x4) = x4 + 2x3 + 6x2 + 2x + 5
(-1i2 + -2i2x + -5i2x2 + 2i3x + -2i3x + 4i4 + 1x2 + 1x3 + 1x3 + x4) = x4 + 2x3 + 6x2 + 2x + 5

Combine like terms: 2i3x + -2i3x = 0
(-1i2 + -2i2x + -5i2x2 + 0 + 4i4 + 1x2 + 1x3 + 1x3 + x4) = x4 + 2x3 + 6x2 + 2x + 5
(-1i2 + -2i2x + -5i2x2 + 4i4 + 1x2 + 1x3 + 1x3 + x4) = x4 + 2x3 + 6x2 + 2x + 5

Combine like terms: 1x3 + 1x3 = 2x3
(-1i2 + -2i2x + -5i2x2 + 4i4 + 1x2 + 2x3 + x4) = x4 + 2x3 + 6x2 + 2x + 5

Reorder the terms:
-1i2 + -2i2x + -5i2x2 + 4i4 + 1x2 + 2x3 + x4 = 5 + 2x + 6x2 + 2x3 + x4

Add '-2x3' to each side of the equation.
-1i2 + -2i2x + -5i2x2 + 4i4 + 1x2 + 2x3 + -2x3 + x4 = 5 + 2x + 6x2 + 2x3 + -2x3 + x4

Combine like terms: 2x3 + -2x3 = 0
-1i2 + -2i2x + -5i2x2 + 4i4 + 1x2 + 0 + x4 = 5 + 2x + 6x2 + 2x3 + -2x3 + x4
-1i2 + -2i2x + -5i2x2 + 4i4 + 1x2 + x4 = 5 + 2x + 6x2 + 2x3 + -2x3 + x4

Combine like terms: 2x3 + -2x3 = 0
-1i2 + -2i2x + -5i2x2 + 4i4 + 1x2 + x4 = 5 + 2x + 6x2 + 0 + x4
-1i2 + -2i2x + -5i2x2 + 4i4 + 1x2 + x4 = 5 + 2x + 6x2 + x4

Add '-1x4' to each side of the equation.
-1i2 + -2i2x + -5i2x2 + 4i4 + 1x2 + x4 + -1x4 = 5 + 2x + 6x2 + x4 + -1x4

Combine like terms: x4 + -1x4 = 0
-1i2 + -2i2x + -5i2x2 + 4i4 + 1x2 + 0 = 5 + 2x + 6x2 + x4 + -1x4
-1i2 + -2i2x + -5i2x2 + 4i4 + 1x2 = 5 + 2x + 6x2 + x4 + -1x4

Combine like terms: x4 + -1x4 = 0
-1i2 + -2i2x + -5i2x2 + 4i4 + 1x2 = 5 + 2x + 6x2 + 0
-1i2 + -2i2x + -5i2x2 + 4i4 + 1x2 = 5 + 2x + 6x2

Solving
-1i2 + -2i2x + -5i2x2 + 4i4 + 1x2 = 5 + 2x + 6x2

Solving for variable 'i'.

Reorder the terms:
-5 + -1i2 + -2i2x + -5i2x2 + 4i4 + -2x + 1x2 + -6x2 = 5 + 2x + 6x2 + -5 + -2x + -6x2

Combine like terms: 1x2 + -6x2 = -5x2
-5 + -1i2 + -2i2x + -5i2x2 + 4i4 + -2x + -5x2 = 5 + 2x + 6x2 + -5 + -2x + -6x2

Reorder the terms:
-5 + -1i2 + -2i2x + -5i2x2 + 4i4 + -2x + -5x2 = 5 + -5 + 2x + -2x + 6x2 + -6x2

Combine like terms: 5 + -5 = 0
-5 + -1i2 + -2i2x + -5i2x2 + 4i4 + -2x + -5x2 = 0 + 2x + -2x + 6x2 + -6x2
-5 + -1i2 + -2i2x + -5i2x2 + 4i4 + -2x + -5x2 = 2x + -2x + 6x2 + -6x2

Combine like terms: 2x + -2x = 0
-5 + -1i2 + -2i2x + -5i2x2 + 4i4 + -2x + -5x2 = 0 + 6x2 + -6x2
-5 + -1i2 + -2i2x + -5i2x2 + 4i4 + -2x + -5x2 = 6x2 + -6x2

Combine like terms: 6x2 + -6x2 = 0
-5 + -1i2 + -2i2x + -5i2x2 + 4i4 + -2x + -5x2 = 0

The solution to this equation could not be determined.

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