Solve (2p-q)(2p+3q)-(p+q)(p-5q)

$$( 2 p - q ) ( 2 p + 3 q ) - ( p + q ) ( p - 5 q )$$

Evaluate

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

Solution Steps

Apply the distributive property by multiplying each term of 2p-q by each term of 2p+3q.

$$4p^{2}+6pq-2qp-3q^{2}-\left(p+q\right)\left(p-5q\right)$$

Combine 6pq and -2qp to get 4pq.

$$4p^{2}+4pq-3q^{2}-\left(p+q\right)\left(p-5q\right)$$

Apply the distributive property by multiplying each term of p+q by each term of p-5q.

$$4p^{2}+4pq-3q^{2}-\left(p^{2}-5pq+qp-5q^{2}\right)$$

Combine -5pq and qp to get -4pq.

$$4p^{2}+4pq-3q^{2}-\left(p^{2}-4pq-5q^{2}\right)$$

To find the opposite of p^{2}-4pq-5q^{2}, find the opposite of each term.

$$4p^{2}+4pq-3q^{2}-p^{2}-\left(-4pq\right)-\left(-5q^{2}\right)$$

The opposite of -4pq is 4pq.

$$4p^{2}+4pq-3q^{2}-p^{2}+4pq-\left(-5q^{2}\right)$$

The opposite of -5q^{2} is 5q^{2}.

$$4p^{2}+4pq-3q^{2}-p^{2}+4pq+5q^{2}$$

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

$$3p^{2}+4pq-3q^{2}+4pq+5q^{2}$$

Combine 4pq and 4pq to get 8pq.

$$3p^{2}+8pq-3q^{2}+5q^{2}$$

Combine -3q^{2} and 5q^{2} to get 2q^{2}.

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

Expand

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

Solution Steps

Apply the distributive property by multiplying each term of 2p-q by each term of 2p+3q.

$$4p^{2}+6pq-2qp-3q^{2}-\left(p+q\right)\left(p-5q\right)$$

Combine 6pq and -2qp to get 4pq.

$$4p^{2}+4pq-3q^{2}-\left(p+q\right)\left(p-5q\right)$$

Apply the distributive property by multiplying each term of p+q by each term of p-5q.

$$4p^{2}+4pq-3q^{2}-\left(p^{2}-5pq+qp-5q^{2}\right)$$

Combine -5pq and qp to get -4pq.

$$4p^{2}+4pq-3q^{2}-\left(p^{2}-4pq-5q^{2}\right)$$

To find the opposite of p^{2}-4pq-5q^{2}, find the opposite of each term.

$$4p^{2}+4pq-3q^{2}-p^{2}-\left(-4pq\right)-\left(-5q^{2}\right)$$

The opposite of -4pq is 4pq.

$$4p^{2}+4pq-3q^{2}-p^{2}+4pq-\left(-5q^{2}\right)$$

The opposite of -5q^{2} is 5q^{2}.

$$4p^{2}+4pq-3q^{2}-p^{2}+4pq+5q^{2}$$

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

$$3p^{2}+4pq-3q^{2}+4pq+5q^{2}$$

Combine 4pq and 4pq to get 8pq.

$$3p^{2}+8pq-3q^{2}+5q^{2}$$

Combine -3q^{2} and 5q^{2} to get 2q^{2}.

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