Solve (p-2q)^2-(2p+q)^2

$$( p - 2 q ) ^ { 2 } - ( 2 p + q ) ^ { 2 }$$

Evaluate

$$-\left(3p-q\right)\left(p+3q\right)$$

Solution Steps

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

$$p^{2}-4pq+4q^{2}-\left(2p+q\right)^{2}$$

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

$$p^{2}-4pq+4q^{2}-\left(4p^{2}+4pq+q^{2}\right)$$

To find the opposite of 4p^{2}+4pq+q^{2}, find the opposite of each term.

$$p^{2}-4pq+4q^{2}-4p^{2}-4pq-q^{2}$$

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

$$-3p^{2}-4pq+4q^{2}-4pq-q^{2}$$

Combine -4pq and -4pq to get -8pq.

$$-3p^{2}-8pq+4q^{2}-q^{2}$$

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

$$-3p^{2}-8pq+3q^{2}$$

Expand

$$3q^{2}-8pq-3p^{2}$$

Solution Steps

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

$$p^{2}-4pq+4q^{2}-\left(2p+q\right)^{2}$$

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

$$p^{2}-4pq+4q^{2}-\left(4p^{2}+4pq+q^{2}\right)$$

To find the opposite of 4p^{2}+4pq+q^{2}, find the opposite of each term.

$$p^{2}-4pq+4q^{2}-4p^{2}-4pq-q^{2}$$

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

$$-3p^{2}-4pq+4q^{2}-4pq-q^{2}$$

Combine -4pq and -4pq to get -8pq.

$$-3p^{2}-8pq+4q^{2}-q^{2}$$

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

$$-3p^{2}-8pq+3q^{2}$$