Solve (5)(m+n)^2-(m+n)^2

$$( 5 ) ( m + n ) ^ { 2 } - ( m + n ) ^ { 2 }$$

Evaluate

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

Solution Steps

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

$$5\left(m^{2}+2mn+n^{2}\right)-\left(m+n\right)^{2}$$

Use the distributive property to multiply 5 by m^{2}+2mn+n^{2}.

$$5m^{2}+10mn+5n^{2}-\left(m+n\right)^{2}$$

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

$$5m^{2}+10mn+5n^{2}-\left(m^{2}+2mn+n^{2}\right)$$

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

$$5m^{2}+10mn+5n^{2}-m^{2}-2mn-n^{2}$$

Combine 5m^{2} and -m^{2} to get 4m^{2}.

$$4m^{2}+10mn+5n^{2}-2mn-n^{2}$$

Combine 10mn and -2mn to get 8mn.

$$4m^{2}+8mn+5n^{2}-n^{2}$$

Combine 5n^{2} and -n^{2} to get 4n^{2}.

$$4m^{2}+8mn+4n^{2}$$

Expand

$$4m^{2}+8mn+4n^{2}$$

Solution Steps

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

$$5\left(m^{2}+2mn+n^{2}\right)-\left(m+n\right)^{2}$$

Use the distributive property to multiply 5 by m^{2}+2mn+n^{2}.

$$5m^{2}+10mn+5n^{2}-\left(m+n\right)^{2}$$

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

$$5m^{2}+10mn+5n^{2}-\left(m^{2}+2mn+n^{2}\right)$$

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

$$5m^{2}+10mn+5n^{2}-m^{2}-2mn-n^{2}$$

Combine 5m^{2} and -m^{2} to get 4m^{2}.

$$4m^{2}+10mn+5n^{2}-2mn-n^{2}$$

Combine 10mn and -2mn to get 8mn.

$$4m^{2}+8mn+5n^{2}-n^{2}$$

Combine 5n^{2} and -n^{2} to get 4n^{2}.

$$4m^{2}+8mn+4n^{2}$$