Solve [(x+2y)^2-(x+y)(3x-y)-5y^2]=2x

$$[ ( x + 2 y ) ^ { 2 } - ( x + y ) ( 3 x - y ) - 5 y ^ { 2 } ] = 2 x$$

Solve for y (complex solution)

$$\left\{\begin{matrix}\\y=x+1\text{, }&\text{unconditionally}\\y\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2y\right)^{2}.

$$x^{2}+4xy+4y^{2}-\left(x+y\right)\left(3x-y\right)-5y^{2}=2x$$

Use the distributive property to multiply x+y by 3x-y and combine like terms.

$$x^{2}+4xy+4y^{2}-\left(3x^{2}+2xy-y^{2}\right)-5y^{2}=2x$$

To find the opposite of 3x^{2}+2xy-y^{2}, find the opposite of each term.

$$x^{2}+4xy+4y^{2}-3x^{2}-2xy+y^{2}-5y^{2}=2x$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$-2x^{2}+4xy+4y^{2}-2xy+y^{2}-5y^{2}=2x$$

Combine 4xy and -2xy to get 2xy.

$$-2x^{2}+2xy+4y^{2}+y^{2}-5y^{2}=2x$$

Combine 4y^{2} and y^{2} to get 5y^{2}.

$$-2x^{2}+2xy+5y^{2}-5y^{2}=2x$$

Combine 5y^{2} and -5y^{2} to get 0.

$$-2x^{2}+2xy=2x$$

Add 2x^{2} to both sides.

$$2xy=2x+2x^{2}$$

The equation is in standard form.

$$2xy=2x^{2}+2x$$

Divide both sides by 2x.

$$\frac{2xy}{2x}=\frac{2x\left(x+1\right)}{2x}$$

Dividing by 2x undoes the multiplication by 2x.

$$y=\frac{2x\left(x+1\right)}{2x}$$

Divide 2x\left(1+x\right) by 2x.

$$y=x+1$$

Solve for y

$$\left\{\begin{matrix}\\y=x+1\text{, }&\text{unconditionally}\\y\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2y\right)^{2}.

$$x^{2}+4xy+4y^{2}-\left(x+y\right)\left(3x-y\right)-5y^{2}=2x$$

Use the distributive property to multiply x+y by 3x-y and combine like terms.

$$x^{2}+4xy+4y^{2}-\left(3x^{2}+2xy-y^{2}\right)-5y^{2}=2x$$

To find the opposite of 3x^{2}+2xy-y^{2}, find the opposite of each term.

$$x^{2}+4xy+4y^{2}-3x^{2}-2xy+y^{2}-5y^{2}=2x$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$-2x^{2}+4xy+4y^{2}-2xy+y^{2}-5y^{2}=2x$$

Combine 4xy and -2xy to get 2xy.

$$-2x^{2}+2xy+4y^{2}+y^{2}-5y^{2}=2x$$

Combine 4y^{2} and y^{2} to get 5y^{2}.

$$-2x^{2}+2xy+5y^{2}-5y^{2}=2x$$

Combine 5y^{2} and -5y^{2} to get 0.

$$-2x^{2}+2xy=2x$$

Add 2x^{2} to both sides.

$$2xy=2x+2x^{2}$$

The equation is in standard form.

$$2xy=2x^{2}+2x$$

Divide both sides by 2x.

$$\frac{2xy}{2x}=\frac{2x\left(x+1\right)}{2x}$$

Dividing by 2x undoes the multiplication by 2x.

$$y=\frac{2x\left(x+1\right)}{2x}$$

Divide 2x\left(1+x\right) by 2x.

$$y=x+1$$

Solve for x

$$x=y-1$$

$$x=0$$