Solve (a+b)^2-(4a-5b)^2

$$( a + b ) ^ { 2 } - ( 4 a - 5 b ) ^ { 2 }$$

Evaluate

$$3\left(2b-a\right)\left(5a-4b\right)$$

Solution Steps

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

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

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(4a-5b\right)^{2}.

$$a^{2}+2ab+b^{2}-\left(16a^{2}-40ab+25b^{2}\right)$$

To find the opposite of 16a^{2}-40ab+25b^{2}, find the opposite of each term.

$$a^{2}+2ab+b^{2}-16a^{2}+40ab-25b^{2}$$

Combine a^{2} and -16a^{2} to get -15a^{2}.

$$-15a^{2}+2ab+b^{2}+40ab-25b^{2}$$

Combine 2ab and 40ab to get 42ab.

$$-15a^{2}+42ab+b^{2}-25b^{2}$$

Combine b^{2} and -25b^{2} to get -24b^{2}.

$$-15a^{2}+42ab-24b^{2}$$

Expand

$$-15a^{2}+42ab-24b^{2}$$

Solution Steps

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

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

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(4a-5b\right)^{2}.

$$a^{2}+2ab+b^{2}-\left(16a^{2}-40ab+25b^{2}\right)$$

To find the opposite of 16a^{2}-40ab+25b^{2}, find the opposite of each term.

$$a^{2}+2ab+b^{2}-16a^{2}+40ab-25b^{2}$$

Combine a^{2} and -16a^{2} to get -15a^{2}.

$$-15a^{2}+2ab+b^{2}+40ab-25b^{2}$$

Combine 2ab and 40ab to get 42ab.

$$-15a^{2}+42ab+b^{2}-25b^{2}$$

Combine b^{2} and -25b^{2} to get -24b^{2}.

$$-15a^{2}+42ab-24b^{2}$$