Solve [(m+n)(m-n)+(m-n)^2-4m(m-n)]=2m

$$[ ( m + n ) ( m - n ) + ( m - n ) ^ { 2 } - 4 m ( m - n ) ] = 2 m$$

Solve for n (complex solution)

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

Steps for Solving Linear Equation

Consider \left(m+n\right)\left(m-n\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$m^{2}-n^{2}+\left(m-n\right)^{2}-4m\left(m-n\right)=2m$$

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

$$m^{2}-n^{2}+m^{2}-2mn+n^{2}-4m\left(m-n\right)=2m$$

Combine m^{2} and m^{2} to get 2m^{2}.

$$2m^{2}-n^{2}-2mn+n^{2}-4m\left(m-n\right)=2m$$

Combine -n^{2} and n^{2} to get 0.

$$2m^{2}-2mn-4m\left(m-n\right)=2m$$

Use the distributive property to multiply -4m by m-n.

$$2m^{2}-2mn-4m^{2}+4mn=2m$$

Combine 2m^{2} and -4m^{2} to get -2m^{2}.

$$-2m^{2}-2mn+4mn=2m$$

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

$$-2m^{2}+2mn=2m$$

Add 2m^{2} to both sides.

$$2mn=2m+2m^{2}$$

The equation is in standard form.

$$2mn=2m^{2}+2m$$

Divide both sides by 2m.

$$\frac{2mn}{2m}=\frac{2m\left(m+1\right)}{2m}$$

Dividing by 2m undoes the multiplication by 2m.

$$n=\frac{2m\left(m+1\right)}{2m}$$

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

$$n=m+1$$

Solve for n

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

Steps for Solving Linear Equation

Consider \left(m+n\right)\left(m-n\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$m^{2}-n^{2}+\left(m-n\right)^{2}-4m\left(m-n\right)=2m$$

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

$$m^{2}-n^{2}+m^{2}-2mn+n^{2}-4m\left(m-n\right)=2m$$

Combine m^{2} and m^{2} to get 2m^{2}.

$$2m^{2}-n^{2}-2mn+n^{2}-4m\left(m-n\right)=2m$$

Combine -n^{2} and n^{2} to get 0.

$$2m^{2}-2mn-4m\left(m-n\right)=2m$$

Use the distributive property to multiply -4m by m-n.

$$2m^{2}-2mn-4m^{2}+4mn=2m$$

Combine 2m^{2} and -4m^{2} to get -2m^{2}.

$$-2m^{2}-2mn+4mn=2m$$

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

$$-2m^{2}+2mn=2m$$

Add 2m^{2} to both sides.

$$2mn=2m+2m^{2}$$

The equation is in standard form.

$$2mn=2m^{2}+2m$$

Divide both sides by 2m.

$$\frac{2mn}{2m}=\frac{2m\left(m+1\right)}{2m}$$

Dividing by 2m undoes the multiplication by 2m.

$$n=\frac{2m\left(m+1\right)}{2m}$$

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

$$n=m+1$$

Solve for m

$$m=n-1$$

$$m=0$$