Solve (3)(3m-4n)(4n+3m)-(2m-n)(2m+n)

$$( 3 ) ( 3 m - 4 n ) ( 4 n + 3 m ) - ( 2 m - n ) ( 2 m + n )$$

Evaluate

$$23m^{2}-47n^{2}$$

Solution Steps

Use the distributive property to multiply 3 by 3m-4n.

$$\left(9m-12n\right)\left(4n+3m\right)-\left(2m-n\right)\left(2m+n\right)$$

Apply the distributive property by multiplying each term of 9m-12n by each term of 4n+3m.

$$36mn+27m^{2}-48n^{2}-36nm-\left(2m-n\right)\left(2m+n\right)$$

Combine 36mn and -36nm to get 0.

$$27m^{2}-48n^{2}-\left(2m-n\right)\left(2m+n\right)$$

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

$$27m^{2}-48n^{2}-\left(\left(2m\right)^{2}-n^{2}\right)$$

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

$$27m^{2}-48n^{2}-\left(2^{2}m^{2}-n^{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$27m^{2}-48n^{2}-\left(4m^{2}-n^{2}\right)$$

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

$$27m^{2}-48n^{2}-4m^{2}-\left(-n^{2}\right)$$

The opposite of -n^{2} is n^{2}.

$$27m^{2}-48n^{2}-4m^{2}+n^{2}$$

Combine 27m^{2} and -4m^{2} to get 23m^{2}.

$$23m^{2}-48n^{2}+n^{2}$$

Combine -48n^{2} and n^{2} to get -47n^{2}.

$$23m^{2}-47n^{2}$$

Expand

$$23m^{2}-47n^{2}$$

Solution Steps

Use the distributive property to multiply 3 by 3m-4n.

$$\left(9m-12n\right)\left(4n+3m\right)-\left(2m-n\right)\left(2m+n\right)$$

Apply the distributive property by multiplying each term of 9m-12n by each term of 4n+3m.

$$36mn+27m^{2}-48n^{2}-36nm-\left(2m-n\right)\left(2m+n\right)$$

Combine 36mn and -36nm to get 0.

$$27m^{2}-48n^{2}-\left(2m-n\right)\left(2m+n\right)$$

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

$$27m^{2}-48n^{2}-\left(\left(2m\right)^{2}-n^{2}\right)$$

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

$$27m^{2}-48n^{2}-\left(2^{2}m^{2}-n^{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$27m^{2}-48n^{2}-\left(4m^{2}-n^{2}\right)$$

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

$$27m^{2}-48n^{2}-4m^{2}-\left(-n^{2}\right)$$

The opposite of -n^{2} is n^{2}.

$$27m^{2}-48n^{2}-4m^{2}+n^{2}$$

Combine 27m^{2} and -4m^{2} to get 23m^{2}.

$$23m^{2}-48n^{2}+n^{2}$$

Combine -48n^{2} and n^{2} to get -47n^{2}.

$$23m^{2}-47n^{2}$$