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

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

Evaluate

$$6m^{2}-4mn-4n^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 4m+2n by 2m-n and combine like terms.

$$8m^{2}-2n^{2}-2\left(m+n\right)^{2}$$

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

$$8m^{2}-2n^{2}-2\left(m^{2}+2mn+n^{2}\right)$$

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

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

Combine 8m^{2} and -2m^{2} to get 6m^{2}.

$$6m^{2}-2n^{2}-4mn-2n^{2}$$

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

$$6m^{2}-4n^{2}-4mn$$

Expand

$$6m^{2}-4mn-4n^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 4m+2n by 2m-n and combine like terms.

$$8m^{2}-2n^{2}-2\left(m+n\right)^{2}$$

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

$$8m^{2}-2n^{2}-2\left(m^{2}+2mn+n^{2}\right)$$

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

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

Combine 8m^{2} and -2m^{2} to get 6m^{2}.

$$6m^{2}-2n^{2}-4mn-2n^{2}$$

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

$$6m^{2}-4n^{2}-4mn$$