Solve 4(p+4)^2-(7p+2q)^2

$$4 ( p + 4 ) ^ { 2 } - ( 7 p + 2 q ) ^ { 2 }$$

Evaluate

$$64-4q^{2}+32p-28pq-45p^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(p+4\right)^{2}.

$$4\left(p^{2}+8p+16\right)-\left(7p+2q\right)^{2}$$

Use the distributive property to multiply 4 by p^{2}+8p+16.

$$4p^{2}+32p+64-\left(7p+2q\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(7p+2q\right)^{2}.

$$4p^{2}+32p+64-\left(49p^{2}+28pq+4q^{2}\right)$$

To find the opposite of 49p^{2}+28pq+4q^{2}, find the opposite of each term.

$$4p^{2}+32p+64-49p^{2}-28pq-4q^{2}$$

Combine 4p^{2} and -49p^{2} to get -45p^{2}.

$$-45p^{2}+32p+64-28pq-4q^{2}$$

Expand

$$64-4q^{2}+32p-28pq-45p^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(p+4\right)^{2}.

$$4\left(p^{2}+8p+16\right)-\left(7p+2q\right)^{2}$$

Use the distributive property to multiply 4 by p^{2}+8p+16.

$$4p^{2}+32p+64-\left(7p+2q\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(7p+2q\right)^{2}.

$$4p^{2}+32p+64-\left(49p^{2}+28pq+4q^{2}\right)$$

To find the opposite of 49p^{2}+28pq+4q^{2}, find the opposite of each term.

$$4p^{2}+32p+64-49p^{2}-28pq-4q^{2}$$

Combine 4p^{2} and -49p^{2} to get -45p^{2}.

$$-45p^{2}+32p+64-28pq-4q^{2}$$