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

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

Evaluate

$$-b\left(4a+5b\right)$$

Solution Steps

Consider \left(a+b\right)\left(a-b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

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

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

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

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

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

$$-5b^{2}-4ab$$

Expand

$$-4ab-5b^{2}$$

Solution Steps

Consider \left(a+b\right)\left(a-b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

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

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

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

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

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

$$-5b^{2}-4ab$$