f(x)=(x-1)(x-1)(x+2i)(x-2i)

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Solution for f(x)=(x-1)(x-1)(x+2i)(x-2i) equation: 


Simplifying
f(x) = (x + -1)(x + -1)(x + 2i)(x + -2i)

Multiply f * x
fx = (x + -1)(x + -1)(x + 2i)(x + -2i)

Reorder the terms:
fx = (-1 + x)(x + -1)(x + 2i)(x + -2i)

Reorder the terms:
fx = (-1 + x)(-1 + x)(x + 2i)(x + -2i)

Reorder the terms:
fx = (-1 + x)(-1 + x)(2i + x)(x + -2i)

Reorder the terms:
fx = (-1 + x)(-1 + x)(2i + x)(-2i + x)

Multiply (-1 + x) * (-1 + x)
fx = (-1(-1 + x) + x(-1 + x))(2i + x)(-2i + x)
fx = ((-1 * -1 + x * -1) + x(-1 + x))(2i + x)(-2i + x)
fx = ((1 + -1x) + x(-1 + x))(2i + x)(-2i + x)
fx = (1 + -1x + (-1 * x + x * x))(2i + x)(-2i + x)
fx = (1 + -1x + (-1x + x2))(2i + x)(-2i + x)

Combine like terms: -1x + -1x = -2x
fx = (1 + -2x + x2)(2i + x)(-2i + x)

Multiply (1 + -2x + x2) * (2i + x)
fx = (1(2i + x) + -2x * (2i + x) + x2(2i + x))(-2i + x)
fx = ((2i * 1 + x * 1) + -2x * (2i + x) + x2(2i + x))(-2i + x)
fx = ((2i + 1x) + -2x * (2i + x) + x2(2i + x))(-2i + x)
fx = (2i + 1x + (2i * -2x + x * -2x) + x2(2i + x))(-2i + x)
fx = (2i + 1x + (-4ix + -2x2) + x2(2i + x))(-2i + x)
fx = (2i + 1x + -4ix + -2x2 + (2i * x2 + x * x2))(-2i + x)
fx = (2i + 1x + -4ix + -2x2 + (2ix2 + x3))(-2i + x)

Reorder the terms:
fx = (2i + -4ix + 2ix2 + 1x + -2x2 + x3)(-2i + x)
fx = (2i + -4ix + 2ix2 + 1x + -2x2 + x3)(-2i + x)

Multiply (2i + -4ix + 2ix2 + 1x + -2x2 + x3) * (-2i + x)
fx = (2i * (-2i + x) + -4ix * (-2i + x) + 2ix2 * (-2i + x) + 1x * (-2i + x) + -2x2 * (-2i + x) + x3(-2i + x))
fx = ((-2i * 2i + x * 2i) + -4ix * (-2i + x) + 2ix2 * (-2i + x) + 1x * (-2i + x) + -2x2 * (-2i + x) + x3(-2i + x))

Reorder the terms:
fx = ((2ix + -4i2) + -4ix * (-2i + x) + 2ix2 * (-2i + x) + 1x * (-2i + x) + -2x2 * (-2i + x) + x3(-2i + x))
fx = ((2ix + -4i2) + -4ix * (-2i + x) + 2ix2 * (-2i + x) + 1x * (-2i + x) + -2x2 * (-2i + x) + x3(-2i + x))
fx = (2ix + -4i2 + (-2i * -4ix + x * -4ix) + 2ix2 * (-2i + x) + 1x * (-2i + x) + -2x2 * (-2i + x) + x3(-2i + x))

Reorder the terms:
fx = (2ix + -4i2 + (-4ix2 + 8i2x) + 2ix2 * (-2i + x) + 1x * (-2i + x) + -2x2 * (-2i + x) + x3(-2i + x))
fx = (2ix + -4i2 + (-4ix2 + 8i2x) + 2ix2 * (-2i + x) + 1x * (-2i + x) + -2x2 * (-2i + x) + x3(-2i + x))
fx = (2ix + -4i2 + -4ix2 + 8i2x + (-2i * 2ix2 + x * 2ix2) + 1x * (-2i + x) + -2x2 * (-2i + x) + x3(-2i + x))

Reorder the terms:
fx = (2ix + -4i2 + -4ix2 + 8i2x + (2ix3 + -4i2x2) + 1x * (-2i + x) + -2x2 * (-2i + x) + x3(-2i + x))
fx = (2ix + -4i2 + -4ix2 + 8i2x + (2ix3 + -4i2x2) + 1x * (-2i + x) + -2x2 * (-2i + x) + x3(-2i + x))
fx = (2ix + -4i2 + -4ix2 + 8i2x + 2ix3 + -4i2x2 + (-2i * 1x + x * 1x) + -2x2 * (-2i + x) + x3(-2i + x))
fx = (2ix + -4i2 + -4ix2 + 8i2x + 2ix3 + -4i2x2 + (-2ix + 1x2) + -2x2 * (-2i + x) + x3(-2i + x))
fx = (2ix + -4i2 + -4ix2 + 8i2x + 2ix3 + -4i2x2 + -2ix + 1x2 + (-2i * -2x2 + x * -2x2) + x3(-2i + x))
fx = (2ix + -4i2 + -4ix2 + 8i2x + 2ix3 + -4i2x2 + -2ix + 1x2 + (4ix2 + -2x3) + x3(-2i + x))
fx = (2ix + -4i2 + -4ix2 + 8i2x + 2ix3 + -4i2x2 + -2ix + 1x2 + 4ix2 + -2x3 + (-2i * x3 + x * x3))
fx = (2ix + -4i2 + -4ix2 + 8i2x + 2ix3 + -4i2x2 + -2ix + 1x2 + 4ix2 + -2x3 + (-2ix3 + x4))

Reorder the terms:
fx = (2ix + -2ix + -4ix2 + 4ix2 + 2ix3 + -2ix3 + -4i2 + 8i2x + -4i2x2 + 1x2 + -2x3 + x4)

Combine like terms: 2ix + -2ix = 0
fx = (0 + -4ix2 + 4ix2 + 2ix3 + -2ix3 + -4i2 + 8i2x + -4i2x2 + 1x2 + -2x3 + x4)
fx = (-4ix2 + 4ix2 + 2ix3 + -2ix3 + -4i2 + 8i2x + -4i2x2 + 1x2 + -2x3 + x4)

Combine like terms: -4ix2 + 4ix2 = 0
fx = (0 + 2ix3 + -2ix3 + -4i2 + 8i2x + -4i2x2 + 1x2 + -2x3 + x4)
fx = (2ix3 + -2ix3 + -4i2 + 8i2x + -4i2x2 + 1x2 + -2x3 + x4)

Combine like terms: 2ix3 + -2ix3 = 0
fx = (0 + -4i2 + 8i2x + -4i2x2 + 1x2 + -2x3 + x4)
fx = (-4i2 + 8i2x + -4i2x2 + 1x2 + -2x3 + x4)

Solving
fx = -4i2 + 8i2x + -4i2x2 + 1x2 + -2x3 + x4

Solving for variable 'f'.

Move all terms containing f to the left, all other terms to the right.

Divide each side by 'x'.
f = -4i2x-1 + 8i2 + -4i2x + x + -2x2 + x3

Simplifying
f = -4i2x-1 + 8i2 + -4i2x + x + -2x2 + x3

Reorder the terms:
f = 8i2 + -4i2x-1 + -4i2x + x + -2x2 + x3

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