f(x+y)=f(x)+f(y)+xy(x+y)

Solution for f(x+y)=f(x)+f(y)+xy(x+y) equation:

Simplifying
f(x + y) = f(x) + f(y) + xy(x + y)
(x * f + y * f) = f(x) + f(y) + xy(x + y)
(fx + fy) = f(x) + f(y) + xy(x + y)

Multiply f * x
fx + fy = fx + f(y) + xy(x + y)

Multiply f * y
fx + fy = fx + fy + xy(x + y)
fx + fy = fx + fy + (x * xy + y * xy)

Reorder the terms:
fx + fy = fx + fy + (xy2 + x2y)
fx + fy = fx + fy + (xy2 + x2y)

Add '-1fx' to each side of the equation.
fx + -1fx + fy = fx + fy + xy2 + -1fx + x2y

Combine like terms: fx + -1fx = 0
0 + fy = fx + fy + xy2 + -1fx + x2y
fy = fx + fy + xy2 + -1fx + x2y

Reorder the terms:
fy = fx + -1fx + fy + xy2 + x2y

Combine like terms: fx + -1fx = 0
fy = 0 + fy + xy2 + x2y
fy = fy + xy2 + x2y

Add '-1fy' to each side of the equation.
fy + -1fy = fy + xy2 + -1fy + x2y

Combine like terms: fy + -1fy = 0
0 = fy + xy2 + -1fy + x2y

Reorder the terms:
0 = fy + -1fy + xy2 + x2y

Combine like terms: fy + -1fy = 0
0 = 0 + xy2 + x2y
0 = xy2 + x2y

Solving
0 = xy2 + x2y

Solving for variable 'x'.
Remove the zero:
-1xy2 + -1x2y = xy2 + x2y + -1xy2 + -1x2y

Reorder the terms:
-1xy2 + -1x2y = xy2 + -1xy2 + x2y + -1x2y

Combine like terms: xy2 + -1xy2 = 0
-1xy2 + -1x2y = 0 + x2y + -1x2y
-1xy2 + -1x2y = x2y + -1x2y

Combine like terms: x2y + -1x2y = 0
-1xy2 + -1x2y = 0

Factor out the Greatest Common Factor (GCF), '-1xy'.
-1xy(y + x) = 0

Ignore the factor -1.
Subproblem 1
Set the factor 'xy' equal to zero and attempt to solve:

Simplifying
xy = 0

Solving
xy = 0

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

Simplifying
xy = 0

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Subproblem 2
Set the factor '(y + x)' equal to zero and attempt to solve:

Simplifying
y + x = 0

Reorder the terms:
x + y = 0

Solving
x + y = 0

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

Add '-1y' to each side of the equation.
x + y + -1y = 0 + -1y

Combine like terms: y + -1y = 0
x + 0 = 0 + -1y
x = 0 + -1y
Remove the zero:
x = -1y

Simplifying
x = -1y
Solution
x = {-1y}