Solve (m+n)^2-(2m+n)^2+(m-9n)^2

$$( m + n ) ^ { 2 } - ( 2 m + n ) ^ { 2 } + ( m - 9 n ) ^ { 2 }$$

Evaluate

$$81n^{2}-20mn-2m^{2}$$

Solution Steps

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

$$m^{2}+2mn+n^{2}-\left(2m+n\right)^{2}+\left(m-9n\right)^{2}$$

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

$$m^{2}+2mn+n^{2}-\left(4m^{2}+4mn+n^{2}\right)+\left(m-9n\right)^{2}$$

To find the opposite of 4m^{2}+4mn+n^{2}, find the opposite of each term.

$$m^{2}+2mn+n^{2}-4m^{2}-4mn-n^{2}+\left(m-9n\right)^{2}$$

Combine m^{2} and -4m^{2} to get -3m^{2}.

$$-3m^{2}+2mn+n^{2}-4mn-n^{2}+\left(m-9n\right)^{2}$$

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

$$-3m^{2}-2mn+n^{2}-n^{2}+\left(m-9n\right)^{2}$$

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

$$-3m^{2}-2mn+\left(m-9n\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-9n\right)^{2}.

$$-3m^{2}-2mn+m^{2}-18mn+81n^{2}$$

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

$$-2m^{2}-2mn-18mn+81n^{2}$$

Combine -2mn and -18mn to get -20mn.

$$-2m^{2}-20mn+81n^{2}$$

Expand

$$81n^{2}-20mn-2m^{2}$$

Solution Steps

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

$$m^{2}+2mn+n^{2}-\left(2m+n\right)^{2}+\left(m-9n\right)^{2}$$

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

$$m^{2}+2mn+n^{2}-\left(4m^{2}+4mn+n^{2}\right)+\left(m-9n\right)^{2}$$

To find the opposite of 4m^{2}+4mn+n^{2}, find the opposite of each term.

$$m^{2}+2mn+n^{2}-4m^{2}-4mn-n^{2}+\left(m-9n\right)^{2}$$

Combine m^{2} and -4m^{2} to get -3m^{2}.

$$-3m^{2}+2mn+n^{2}-4mn-n^{2}+\left(m-9n\right)^{2}$$

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

$$-3m^{2}-2mn+n^{2}-n^{2}+\left(m-9n\right)^{2}$$

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

$$-3m^{2}-2mn+\left(m-9n\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-9n\right)^{2}.

$$-3m^{2}-2mn+m^{2}-18mn+81n^{2}$$

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

$$-2m^{2}-2mn-18mn+81n^{2}$$

Combine -2mn and -18mn to get -20mn.

$$-2m^{2}-20mn+81n^{2}$$