(1+x+x^2+x^3)(1-x)=(1+x+x^2+x^3)1-x(1+x+x^2+x^3)

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Solution for (1+x+x^2+x^3)(1-x)=(1+x+x^2+x^3)1-x(1+x+x^2+x^3) equation: 


Simplifying
(1 + x + x2 + x3)(1 + -1x) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)

Multiply (1 + x + x2 + x3) * (1 + -1x)
(1(1 + -1x) + x(1 + -1x) + x2(1 + -1x) + x3(1 + -1x)) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)
((1 * 1 + -1x * 1) + x(1 + -1x) + x2(1 + -1x) + x3(1 + -1x)) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)
((1 + -1x) + x(1 + -1x) + x2(1 + -1x) + x3(1 + -1x)) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)
(1 + -1x + (1 * x + -1x * x) + x2(1 + -1x) + x3(1 + -1x)) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)
(1 + -1x + (1x + -1x2) + x2(1 + -1x) + x3(1 + -1x)) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)
(1 + -1x + 1x + -1x2 + (1 * x2 + -1x * x2) + x3(1 + -1x)) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)
(1 + -1x + 1x + -1x2 + (1x2 + -1x3) + x3(1 + -1x)) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)
(1 + -1x + 1x + -1x2 + 1x2 + -1x3 + (1 * x3 + -1x * x3)) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)
(1 + -1x + 1x + -1x2 + 1x2 + -1x3 + (1x3 + -1x4)) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)

Combine like terms: -1x + 1x = 0
(1 + 0 + -1x2 + 1x2 + -1x3 + 1x3 + -1x4) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)
(1 + -1x2 + 1x2 + -1x3 + 1x3 + -1x4) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)

Combine like terms: -1x2 + 1x2 = 0
(1 + 0 + -1x3 + 1x3 + -1x4) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)
(1 + -1x3 + 1x3 + -1x4) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)

Combine like terms: -1x3 + 1x3 = 0
(1 + 0 + -1x4) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)
(1 + -1x4) = (1 + x + x2 + x3) * 1 + -1x(1 + x + x2 + x3)

Reorder the terms for easier multiplication:
1 + -1x4 = 1(1 + x + x2 + x3) + -1x(1 + x + x2 + x3)
1 + -1x4 = (1 * 1 + x * 1 + x2 * 1 + x3 * 1) + -1x(1 + x + x2 + x3)
1 + -1x4 = (1 + 1x + 1x2 + 1x3) + -1x(1 + x + x2 + x3)
1 + -1x4 = 1 + 1x + 1x2 + 1x3 + (1 * -1x + x * -1x + x2 * -1x + x3 * -1x)
1 + -1x4 = 1 + 1x + 1x2 + 1x3 + (-1x + -1x2 + -1x3 + -1x4)

Reorder the terms:
1 + -1x4 = 1 + 1x + -1x + 1x2 + -1x2 + 1x3 + -1x3 + -1x4

Combine like terms: 1x + -1x = 0
1 + -1x4 = 1 + 0 + 1x2 + -1x2 + 1x3 + -1x3 + -1x4
1 + -1x4 = 1 + 1x2 + -1x2 + 1x3 + -1x3 + -1x4

Combine like terms: 1x2 + -1x2 = 0
1 + -1x4 = 1 + 0 + 1x3 + -1x3 + -1x4
1 + -1x4 = 1 + 1x3 + -1x3 + -1x4

Combine like terms: 1x3 + -1x3 = 0
1 + -1x4 = 1 + 0 + -1x4
1 + -1x4 = 1 + -1x4

Add '-1' to each side of the equation.
1 + -1 + -1x4 = 1 + -1 + -1x4

Combine like terms: 1 + -1 = 0
0 + -1x4 = 1 + -1 + -1x4
-1x4 = 1 + -1 + -1x4

Combine like terms: 1 + -1 = 0
-1x4 = 0 + -1x4
-1x4 = -1x4

Add 'x4' to each side of the equation.
-1x4 + x4 = -1x4 + x4

Combine like terms: -1x4 + x4 = 0
0 = -1x4 + x4

Combine like terms: -1x4 + x4 = 0
0 = 0

Solving
0 = 0

Couldn't find a variable to solve for.

This equation is an identity, all real numbers are solutions.

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