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

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

Evaluate

$$8ab$$

Solution Steps

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

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

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

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

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

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

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

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

Combine 4ab and 4ba to get 8ab.

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

Combine 4a^{2} and -4a^{2} to get 0.

$$8ab$$

Expand

$$8ab$$

Solution Steps

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

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

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

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

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

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

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

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

Combine 4ab and 4ba to get 8ab.

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

Combine 4a^{2} and -4a^{2} to get 0.

$$8ab$$