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

$$f ( x ) + f ( 2 + y ) + 5 x y = f ( 3 x - y ) + x ^ { 2 } + 1$$

Solve for f (complex solution)

$$\left\{\begin{matrix}f=-\frac{-x^{2}+5xy-1}{2\left(1+y-x\right)}\text{, }&x\neq y+1\\f\in \mathrm{C}\text{, }&\left(x=\frac{5-\sqrt{41}}{8}\text{ and }y=\frac{-\sqrt{41}-3}{8}\right)\text{ or }\left(x=\frac{\sqrt{41}+5}{8}\text{ and }y=\frac{\sqrt{41}-3}{8}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply f by 2+y.

$$fx+2f+fy+5xy=f\left(3x-y\right)+x^{2}+1$$

Use the distributive property to multiply f by 3x-y.

$$fx+2f+fy+5xy=3fx-fy+x^{2}+1$$

Subtract 3fx from both sides.

$$fx+2f+fy+5xy-3fx=-fy+x^{2}+1$$

Combine fx and -3fx to get -2fx.

$$-2fx+2f+fy+5xy=-fy+x^{2}+1$$

Add fy to both sides.

$$-2fx+2f+fy+5xy+fy=x^{2}+1$$

Combine fy and fy to get 2fy.

$$-2fx+2f+2fy+5xy=x^{2}+1$$

Subtract 5xy from both sides.

$$-2fx+2f+2fy=x^{2}+1-5xy$$

Combine all terms containing f.

$$\left(-2x+2+2y\right)f=x^{2}+1-5xy$$

The equation is in standard form.

$$\left(2+2y-2x\right)f=x^{2}-5xy+1$$

Divide both sides by 2+2y-2x.

$$\frac{\left(2+2y-2x\right)f}{2+2y-2x}=\frac{x^{2}-5xy+1}{2+2y-2x}$$

Dividing by 2+2y-2x undoes the multiplication by 2+2y-2x.

$$f=\frac{x^{2}-5xy+1}{2+2y-2x}$$

Divide x^{2}+1-5xy by 2+2y-2x.

$$f=\frac{x^{2}-5xy+1}{2\left(1+y-x\right)}$$

Solve for f

$$\left\{\begin{matrix}f=-\frac{-x^{2}+5xy-1}{2\left(1+y-x\right)}\text{, }&x\neq y+1\\f\in \mathrm{R}\text{, }&\left(x=\frac{5-\sqrt{41}}{8}\text{ and }y=\frac{-\sqrt{41}-3}{8}\right)\text{ or }\left(x=\frac{\sqrt{41}+5}{8}\text{ and }y=\frac{\sqrt{41}-3}{8}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply f by 2+y.

$$fx+2f+fy+5xy=f\left(3x-y\right)+x^{2}+1$$

Use the distributive property to multiply f by 3x-y.

$$fx+2f+fy+5xy=3fx-fy+x^{2}+1$$

Subtract 3fx from both sides.

$$fx+2f+fy+5xy-3fx=-fy+x^{2}+1$$

Combine fx and -3fx to get -2fx.

$$-2fx+2f+fy+5xy=-fy+x^{2}+1$$

Add fy to both sides.

$$-2fx+2f+fy+5xy+fy=x^{2}+1$$

Combine fy and fy to get 2fy.

$$-2fx+2f+2fy+5xy=x^{2}+1$$

Subtract 5xy from both sides.

$$-2fx+2f+2fy=x^{2}+1-5xy$$

Combine all terms containing f.

$$\left(-2x+2+2y\right)f=x^{2}+1-5xy$$

The equation is in standard form.

$$\left(2+2y-2x\right)f=x^{2}-5xy+1$$

Divide both sides by 2+2y-2x.

$$\frac{\left(2+2y-2x\right)f}{2+2y-2x}=\frac{x^{2}-5xy+1}{2+2y-2x}$$

Dividing by 2+2y-2x undoes the multiplication by 2+2y-2x.

$$f=\frac{x^{2}-5xy+1}{2+2y-2x}$$

Divide x^{2}+1-5xy by 2+2y-2x.

$$f=\frac{x^{2}-5xy+1}{2\left(1+y-x\right)}$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{25y^{2}-12fy+4f^{2}+8f-4}}{2}+\frac{5y}{2}-f$$

$$x=\frac{\sqrt{25y^{2}-12fy+4f^{2}+8f-4}}{2}+\frac{5y}{2}-f$$

Solve for x

$$x=-\frac{\sqrt{25y^{2}-12fy+4f^{2}+8f-4}}{2}+\frac{5y}{2}-f$$

$$x=\frac{\sqrt{25y^{2}-12fy+4f^{2}+8f-4}}{2}+\frac{5y}{2}-f\text{, }f\geq \sqrt{2-3y-4y^{2}}+\frac{3y}{2}-1\text{ or }f\leq -\sqrt{2-3y-4y^{2}}+\frac{3y}{2}-1\text{ or }y\leq \frac{-\sqrt{41}-3}{8}\text{ or }y\geq \frac{\sqrt{41}-3}{8}$$