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

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

Evaluate

$$-4b\left(a+2b\right)$$

Solution Steps

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

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

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+3b\right)^{2}.

$$a^{2}+2ab+b^{2}-\left(a^{2}+6ab+9b^{2}\right)$$

To find the opposite of a^{2}+6ab+9b^{2}, find the opposite of each term.

$$a^{2}+2ab+b^{2}-a^{2}-6ab-9b^{2}$$

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

$$2ab+b^{2}-6ab-9b^{2}$$

Combine 2ab and -6ab to get -4ab.

$$-4ab+b^{2}-9b^{2}$$

Combine b^{2} and -9b^{2} to get -8b^{2}.

$$-4ab-8b^{2}$$

Expand

$$-4ab-8b^{2}$$

Solution Steps

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

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

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+3b\right)^{2}.

$$a^{2}+2ab+b^{2}-\left(a^{2}+6ab+9b^{2}\right)$$

To find the opposite of a^{2}+6ab+9b^{2}, find the opposite of each term.

$$a^{2}+2ab+b^{2}-a^{2}-6ab-9b^{2}$$

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

$$2ab+b^{2}-6ab-9b^{2}$$

Combine 2ab and -6ab to get -4ab.

$$-4ab+b^{2}-9b^{2}$$

Combine b^{2} and -9b^{2} to get -8b^{2}.

$$-4ab-8b^{2}$$