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

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

Evaluate

$$2a+2b-15$$

Solution Steps

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

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

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

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

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

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

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

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

Combine 2ab and -2ab to get 0.

$$2a+b^{2}+2b-15-b^{2}$$

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

$$2a+2b-15$$

Expand

$$2a+2b-15$$

Solution Steps

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

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

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

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

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

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

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

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

Combine 2ab and -2ab to get 0.

$$2a+b^{2}+2b-15-b^{2}$$

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

$$2a+2b-15$$