Solve (a+3b)(a+5b)-(a+4b)^2

$$( a + 3 b ) ( a + 5 b ) - ( a + 4 b ) ^ { 2 }$$

Evaluate

$$-b^{2}$$

Solution Steps

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

$$a^{2}+8ab+15b^{2}-\left(a+4b\right)^{2}$$

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

$$a^{2}+8ab+15b^{2}-\left(a^{2}+8ab+16b^{2}\right)$$

To find the opposite of a^{2}+8ab+16b^{2}, find the opposite of each term.

$$a^{2}+8ab+15b^{2}-a^{2}-8ab-16b^{2}$$

Combine a^{2} and -a^{2} to get 0.

$$8ab+15b^{2}-8ab-16b^{2}$$

Combine 8ab and -8ab to get 0.

$$15b^{2}-16b^{2}$$

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

$$-b^{2}$$

Expand

$$-b^{2}$$

Solution Steps

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

$$a^{2}+8ab+15b^{2}-\left(a+4b\right)^{2}$$

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

$$a^{2}+8ab+15b^{2}-\left(a^{2}+8ab+16b^{2}\right)$$

To find the opposite of a^{2}+8ab+16b^{2}, find the opposite of each term.

$$a^{2}+8ab+15b^{2}-a^{2}-8ab-16b^{2}$$

Combine a^{2} and -a^{2} to get 0.

$$8ab+15b^{2}-8ab-16b^{2}$$

Combine 8ab and -8ab to get 0.

$$15b^{2}-16b^{2}$$

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

$$-b^{2}$$