Solve (m+n)(-2n)-(m-3n)(m+n)+(-6m^3n^2):(+2mn^2)

$$( m + n ) ( - 2 n ) - ( m - 3 n ) ( m + n ) + ( - 6 m ^ { 3 } n ^ { 2 } ) : ( + 2 m n ^ { 2 } )$$

Evaluate

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

Solution Steps

Cancel out 2mn^{2} in both numerator and denominator.

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

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

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

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

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

Use the distributive property to multiply m-3n by m+n and combine like terms.

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

To find the opposite of m^{2}-2mn-3n^{2}, find the opposite of each term.

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

Combine -2mn and 2mn to get 0.

$$-2n^{2}-m^{2}+3n^{2}-3m^{2}$$

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

$$n^{2}-m^{2}-3m^{2}$$

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

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

Expand

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

Solution Steps

Cancel out 2mn^{2} in both numerator and denominator.

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

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

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

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

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

Use the distributive property to multiply m-3n by m+n and combine like terms.

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

To find the opposite of m^{2}-2mn-3n^{2}, find the opposite of each term.

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

Combine -2mn and 2mn to get 0.

$$-2n^{2}-m^{2}+3n^{2}-3m^{2}$$

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

$$n^{2}-m^{2}-3m^{2}$$

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

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