Solve 2(m^2+n^2)-(m-n)^2

$$2 ( m ^ { 2 } + n ^ { 2 } ) - ( m - n ) ^ { 2 }$$

Evaluate

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

Solution Steps

Use the distributive property to multiply 2 by m^{2}+n^{2}.

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

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

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

To find the opposite of m^{2}-2mn+n^{2}, find the opposite of each term.

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

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

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

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

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

Expand

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

Solution Steps

Use the distributive property to multiply 2 by m^{2}+n^{2}.

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

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

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

To find the opposite of m^{2}-2mn+n^{2}, find the opposite of each term.

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

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

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

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

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