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

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

Evaluate

$$\left(a-3b\right)\left(3a+b\right)$$

Solution Steps

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

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

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

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

To find the opposite of a^{2}+4ab+4b^{2}, find the opposite of each term.

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

Combine 4a^{2} and -a^{2} to get 3a^{2}.

$$3a^{2}-4ab+b^{2}-4ab-4b^{2}$$

Combine -4ab and -4ab to get -8ab.

$$3a^{2}-8ab+b^{2}-4b^{2}$$

Combine b^{2} and -4b^{2} to get -3b^{2}.

$$3a^{2}-8ab-3b^{2}$$

Expand

$$3a^{2}-8ab-3b^{2}$$

Solution Steps

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

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

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

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

To find the opposite of a^{2}+4ab+4b^{2}, find the opposite of each term.

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

Combine 4a^{2} and -a^{2} to get 3a^{2}.

$$3a^{2}-4ab+b^{2}-4ab-4b^{2}$$

Combine -4ab and -4ab to get -8ab.

$$3a^{2}-8ab+b^{2}-4b^{2}$$

Combine b^{2} and -4b^{2} to get -3b^{2}.

$$3a^{2}-8ab-3b^{2}$$