Solve (4-x^2)^2-(2-x)^2

$$( 4 - x ^ { 2 } ) ^ { 2 } - ( 2 - x ) ^ { 2 }$$

Evaluate

$$\left(x+1\right)\left(x+3\right)\left(x-2\right)^{2}$$

Solution Steps

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

$$16-8x^{2}+\left(x^{2}\right)^{2}-\left(2-x\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$16-8x^{2}+x^{4}-\left(2-x\right)^{2}$$

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

$$16-8x^{2}+x^{4}-\left(4-4x+x^{2}\right)$$

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

$$16-8x^{2}+x^{4}-4+4x-x^{2}$$

Subtract 4 from 16 to get 12.

$$12-8x^{2}+x^{4}+4x-x^{2}$$

Combine -8x^{2} and -x^{2} to get -9x^{2}.

$$12-9x^{2}+x^{4}+4x$$

Expand

$$x^{4}-9x^{2}+4x+12$$

Solution Steps

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

$$16-8x^{2}+\left(x^{2}\right)^{2}-\left(2-x\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$16-8x^{2}+x^{4}-\left(2-x\right)^{2}$$

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

$$16-8x^{2}+x^{4}-\left(4-4x+x^{2}\right)$$

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

$$16-8x^{2}+x^{4}-4+4x-x^{2}$$

Subtract 4 from 16 to get 12.

$$12-8x^{2}+x^{4}+4x-x^{2}$$

Combine -8x^{2} and -x^{2} to get -9x^{2}.

$$12-9x^{2}+x^{4}+4x$$