Solve left(4x-2right)^2-left(2+4xright)^2

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

Evaluate

$$-32x$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-2\right)^{2}.

$$16x^{2}-16x+4-\left(2+4x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+4x\right)^{2}.

$$16x^{2}-16x+4-\left(4+16x+16x^{2}\right)$$

To find the opposite of 4+16x+16x^{2}, find the opposite of each term.

$$16x^{2}-16x+4-4-16x-16x^{2}$$

Subtract 4 from 4 to get 0.

$$16x^{2}-16x-16x-16x^{2}$$

Combine -16x and -16x to get -32x.

$$16x^{2}-32x-16x^{2}$$

Combine 16x^{2} and -16x^{2} to get 0.

$$-32x$$

Expand

$$-32x$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-2\right)^{2}.

$$16x^{2}-16x+4-\left(2+4x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+4x\right)^{2}.

$$16x^{2}-16x+4-\left(4+16x+16x^{2}\right)$$

To find the opposite of 4+16x+16x^{2}, find the opposite of each term.

$$16x^{2}-16x+4-4-16x-16x^{2}$$

Subtract 4 from 4 to get 0.

$$16x^{2}-16x-16x-16x^{2}$$

Combine -16x and -16x to get -32x.

$$16x^{2}-32x-16x^{2}$$

Combine 16x^{2} and -16x^{2} to get 0.

$$-32x$$