Solve left(7a+2bright)^2-left(3a-bright)left(4a+5bright)

$${ \left(7a+2b \right) }^{ 2 } - \left( 3a-b \right) \left( 4a+5b \right)$$

Evaluate

$$37a^{2}+17ab+9b^{2}$$

Solution Steps

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

$$49a^{2}+28ab+4b^{2}-\left(3a-b\right)\left(4a+5b\right)$$

Use the distributive property to multiply 3a-b by 4a+5b and combine like terms.

$$49a^{2}+28ab+4b^{2}-\left(12a^{2}+11ab-5b^{2}\right)$$

To find the opposite of 12a^{2}+11ab-5b^{2}, find the opposite of each term.

$$49a^{2}+28ab+4b^{2}-12a^{2}-11ab+5b^{2}$$

Combine 49a^{2} and -12a^{2} to get 37a^{2}.

$$37a^{2}+28ab+4b^{2}-11ab+5b^{2}$$

Combine 28ab and -11ab to get 17ab.

$$37a^{2}+17ab+4b^{2}+5b^{2}$$

Combine 4b^{2} and 5b^{2} to get 9b^{2}.

$$37a^{2}+17ab+9b^{2}$$

Expand

$$37a^{2}+17ab+9b^{2}$$

Solution Steps

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

$$49a^{2}+28ab+4b^{2}-\left(3a-b\right)\left(4a+5b\right)$$

Use the distributive property to multiply 3a-b by 4a+5b and combine like terms.

$$49a^{2}+28ab+4b^{2}-\left(12a^{2}+11ab-5b^{2}\right)$$

To find the opposite of 12a^{2}+11ab-5b^{2}, find the opposite of each term.

$$49a^{2}+28ab+4b^{2}-12a^{2}-11ab+5b^{2}$$

Combine 49a^{2} and -12a^{2} to get 37a^{2}.

$$37a^{2}+28ab+4b^{2}-11ab+5b^{2}$$

Combine 28ab and -11ab to get 17ab.

$$37a^{2}+17ab+4b^{2}+5b^{2}$$

Combine 4b^{2} and 5b^{2} to get 9b^{2}.

$$37a^{2}+17ab+9b^{2}$$