Solve (2m+3n)(3n-2m)-2(-3m-2n)(2n-3m)

$$( 2 m + 3 n ) ( 3 n - 2 m ) - 2 ( - 3 m - 2 n ) ( 2 n - 3 m )$$

Evaluate

$$17n^{2}-22m^{2}$$

Solution Steps

Consider \left(2m+3n\right)\left(3n-2m\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(3n\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

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

Expand \left(2m\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

Use the distributive property to multiply -2 by -3m-2n.

$$9n^{2}-4m^{2}+\left(6m+4n\right)\left(2n-3m\right)$$

Apply the distributive property by multiplying each term of 6m+4n by each term of 2n-3m.

$$9n^{2}-4m^{2}+12mn-18m^{2}+8n^{2}-12nm$$

Combine 12mn and -12nm to get 0.

$$9n^{2}-4m^{2}-18m^{2}+8n^{2}$$

Combine -4m^{2} and -18m^{2} to get -22m^{2}.

$$9n^{2}-22m^{2}+8n^{2}$$

Combine 9n^{2} and 8n^{2} to get 17n^{2}.

$$17n^{2}-22m^{2}$$

Expand

$$17n^{2}-22m^{2}$$

Solution Steps

Consider \left(2m+3n\right)\left(3n-2m\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(3n\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

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

Expand \left(2m\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

Use the distributive property to multiply -2 by -3m-2n.

$$9n^{2}-4m^{2}+\left(6m+4n\right)\left(2n-3m\right)$$

Apply the distributive property by multiplying each term of 6m+4n by each term of 2n-3m.

$$9n^{2}-4m^{2}+12mn-18m^{2}+8n^{2}-12nm$$

Combine 12mn and -12nm to get 0.

$$9n^{2}-4m^{2}-18m^{2}+8n^{2}$$

Combine -4m^{2} and -18m^{2} to get -22m^{2}.

$$9n^{2}-22m^{2}+8n^{2}$$

Combine 9n^{2} and 8n^{2} to get 17n^{2}.

$$17n^{2}-22m^{2}$$