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

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

Evaluate

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

Solution Steps

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

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

Use the distributive property to multiply 4 by a^{2}+2ab+b^{2}.

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

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

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

Combine 4a^{2} and 4a^{2} to get 8a^{2}.

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

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

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

Combine 4b^{2} and b^{2} to get 5b^{2}.

$$8a^{2}+4ab+5b^{2}-b^{2}-\left(a+2b\right)$$

Combine 5b^{2} and -b^{2} to get 4b^{2}.

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

To find the opposite of a+2b, find the opposite of each term.

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

Expand

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

Solution Steps

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

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

Use the distributive property to multiply 4 by a^{2}+2ab+b^{2}.

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

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

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

Combine 4a^{2} and 4a^{2} to get 8a^{2}.

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

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

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

Combine 4b^{2} and b^{2} to get 5b^{2}.

$$8a^{2}+4ab+5b^{2}-b^{2}-\left(a+2b\right)$$

Combine 5b^{2} and -b^{2} to get 4b^{2}.

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

To find the opposite of a+2b, find the opposite of each term.

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