Solve 4(2p+3q)^2-(3p-q)^2

$$4 ( 2 p + 3 q ) ^ { 2 } - ( 3 p - q ) ^ { 2 }$$

Evaluate

$$\left(p+7q\right)\left(7p+5q\right)$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2p+3q\right)^{2}.

$$4\left(4p^{2}+12pq+9q^{2}\right)-\left(3p-q\right)^{2}$$

Use the distributive property to multiply 4 by 4p^{2}+12pq+9q^{2}.

$$16p^{2}+48pq+36q^{2}-\left(3p-q\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3p-q\right)^{2}.

$$16p^{2}+48pq+36q^{2}-\left(9p^{2}-6pq+q^{2}\right)$$

To find the opposite of 9p^{2}-6pq+q^{2}, find the opposite of each term.

$$16p^{2}+48pq+36q^{2}-9p^{2}+6pq-q^{2}$$

Combine 16p^{2} and -9p^{2} to get 7p^{2}.

$$7p^{2}+48pq+36q^{2}+6pq-q^{2}$$

Combine 48pq and 6pq to get 54pq.

$$7p^{2}+54pq+36q^{2}-q^{2}$$

Combine 36q^{2} and -q^{2} to get 35q^{2}.

$$7p^{2}+54pq+35q^{2}$$

Expand

$$7p^{2}+54pq+35q^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2p+3q\right)^{2}.

$$4\left(4p^{2}+12pq+9q^{2}\right)-\left(3p-q\right)^{2}$$

Use the distributive property to multiply 4 by 4p^{2}+12pq+9q^{2}.

$$16p^{2}+48pq+36q^{2}-\left(3p-q\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3p-q\right)^{2}.

$$16p^{2}+48pq+36q^{2}-\left(9p^{2}-6pq+q^{2}\right)$$

To find the opposite of 9p^{2}-6pq+q^{2}, find the opposite of each term.

$$16p^{2}+48pq+36q^{2}-9p^{2}+6pq-q^{2}$$

Combine 16p^{2} and -9p^{2} to get 7p^{2}.

$$7p^{2}+48pq+36q^{2}+6pq-q^{2}$$

Combine 48pq and 6pq to get 54pq.

$$7p^{2}+54pq+36q^{2}-q^{2}$$

Combine 36q^{2} and -q^{2} to get 35q^{2}.

$$7p^{2}+54pq+35q^{2}$$