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

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

Evaluate

$$9m^{2}$$

Solution Steps

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

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

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

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

Use the distributive property to multiply 2m+2n by 2m-n and combine like terms.

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

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

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

Combine 2mn and 2mn to get 4mn.

$$5m^{2}+4mn+n^{2}-2n^{2}+\left(2m-n\right)^{2}$$

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

$$5m^{2}+4mn-n^{2}+\left(2m-n\right)^{2}$$

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

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

Combine 5m^{2} and 4m^{2} to get 9m^{2}.

$$9m^{2}+4mn-n^{2}-4mn+n^{2}$$

Combine 4mn and -4mn to get 0.

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

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

$$9m^{2}$$

Expand

$$9m^{2}$$

Solution Steps

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

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

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

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

Use the distributive property to multiply 2m+2n by 2m-n and combine like terms.

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

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

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

Combine 2mn and 2mn to get 4mn.

$$5m^{2}+4mn+n^{2}-2n^{2}+\left(2m-n\right)^{2}$$

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

$$5m^{2}+4mn-n^{2}+\left(2m-n\right)^{2}$$

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

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

Combine 5m^{2} and 4m^{2} to get 9m^{2}.

$$9m^{2}+4mn-n^{2}-4mn+n^{2}$$

Combine 4mn and -4mn to get 0.

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

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

$$9m^{2}$$