f(x)+f(2x+y)+5xy=f(3x-y)+2x^2+1

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Solution for f(x)+f(2x+y)+5xy=f(3x-y)+2x^2+1 equation: 


Simplifying
f(x) + f(2x + y) + 5xy = f(3x + -1y) + 2x2 + 1

Multiply f * x
fx + f(2x + y) + 5xy = f(3x + -1y) + 2x2 + 1
fx + (2x * f + y * f) + 5xy = f(3x + -1y) + 2x2 + 1
fx + (2fx + fy) + 5xy = f(3x + -1y) + 2x2 + 1

Combine like terms: fx + 2fx = 3fx
3fx + fy + 5xy = f(3x + -1y) + 2x2 + 1
3fx + fy + 5xy = (3x * f + -1y * f) + 2x2 + 1
3fx + fy + 5xy = (3fx + -1fy) + 2x2 + 1

Reorder the terms:
3fx + fy + 5xy = 1 + 3fx + -1fy + 2x2

Add '-3fx' to each side of the equation.
3fx + fy + -3fx + 5xy = 1 + 3fx + -1fy + -3fx + 2x2

Reorder the terms:
3fx + -3fx + fy + 5xy = 1 + 3fx + -1fy + -3fx + 2x2

Combine like terms: 3fx + -3fx = 0
0 + fy + 5xy = 1 + 3fx + -1fy + -3fx + 2x2
fy + 5xy = 1 + 3fx + -1fy + -3fx + 2x2

Reorder the terms:
fy + 5xy = 1 + 3fx + -3fx + -1fy + 2x2

Combine like terms: 3fx + -3fx = 0
fy + 5xy = 1 + 0 + -1fy + 2x2
fy + 5xy = 1 + -1fy + 2x2

Solving
fy + 5xy = 1 + -1fy + 2x2

Solving for variable 'f'.

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

Add 'fy' to each side of the equation.
fy + fy + 5xy = 1 + -1fy + fy + 2x2

Combine like terms: fy + fy = 2fy
2fy + 5xy = 1 + -1fy + fy + 2x2

Combine like terms: -1fy + fy = 0
2fy + 5xy = 1 + 0 + 2x2
2fy + 5xy = 1 + 2x2

Add '-5xy' to each side of the equation.
2fy + 5xy + -5xy = 1 + -5xy + 2x2

Combine like terms: 5xy + -5xy = 0
2fy + 0 = 1 + -5xy + 2x2
2fy = 1 + -5xy + 2x2

Divide each side by '2y'.
f = 0.5y-1 + -2.5x + x2y-1

Simplifying
f = 0.5y-1 + -2.5x + x2y-1

Reorder the terms:
f = -2.5x + x2y-1 + 0.5y-1

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