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

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

Evaluate

$$-8ab$$

Solution Steps

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

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

Calculate -a to the power of 2 and get a^{2}.

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

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

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

Calculate -a to the power of 2 and get a^{2}.

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

Multiply -4 and -1 to get 4.

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

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

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

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

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

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

$$4\left(-a\right)b-4ab$$

Multiply 4 and -1 to get -4.

$$-4ab-4ab$$

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

$$-8ab$$

Expand

$$-8ab$$

Solution Steps

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

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

Calculate -a to the power of 2 and get a^{2}.

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

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

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

Calculate -a to the power of 2 and get a^{2}.

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

Multiply -4 and -1 to get 4.

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

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

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

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

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

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

$$4\left(-a\right)b-4ab$$

Multiply 4 and -1 to get -4.

$$-4ab-4ab$$

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

$$-8ab$$