Solve left(-x^2+4right)^2+2(-x^2+4)-3

$${ \left(- { x }^{ 2 } +4 \right) }^{ 2 } +2(- { x }^{ 2 } +4)-3$$

Evaluate

$$\left(x^{2}-7\right)\left(x^{2}-3\right)$$

Solution Steps

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

$$\left(-x^{2}\right)^{2}+8\left(-x^{2}\right)+16+2\left(-x^{2}+4\right)-3$$

Calculate -x^{2} to the power of 2 and get \left(x^{2}\right)^{2}.

$$\left(x^{2}\right)^{2}+8\left(-x^{2}\right)+16+2\left(-x^{2}+4\right)-3$$

Use the distributive property to multiply 2 by -x^{2}+4.

$$\left(x^{2}\right)^{2}+8\left(-x^{2}\right)+16+2\left(-x^{2}\right)+8-3$$

Combine 8\left(-x^{2}\right) and 2\left(-x^{2}\right) to get 10\left(-x^{2}\right).

$$\left(x^{2}\right)^{2}+10\left(-x^{2}\right)+16+8-3$$

Add 16 and 8 to get 24.

$$\left(x^{2}\right)^{2}+10\left(-x^{2}\right)+24-3$$

Subtract 3 from 24 to get 21.

$$\left(x^{2}\right)^{2}+10\left(-x^{2}\right)+21$$

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

$$x^{4}+10\left(-x^{2}\right)+21$$

Multiply 10 and -1 to get -10.

$$x^{4}-10x^{2}+21$$

Expand

$$x^{4}-10x^{2}+21$$

Solution Steps

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

$$\left(-x^{2}\right)^{2}+8\left(-x^{2}\right)+16+2\left(-x^{2}+4\right)-3$$

Calculate -x^{2} to the power of 2 and get \left(x^{2}\right)^{2}.

$$\left(x^{2}\right)^{2}+8\left(-x^{2}\right)+16+2\left(-x^{2}+4\right)-3$$

Use the distributive property to multiply 2 by -x^{2}+4.

$$\left(x^{2}\right)^{2}+8\left(-x^{2}\right)+16+2\left(-x^{2}\right)+8-3$$

Combine 8\left(-x^{2}\right) and 2\left(-x^{2}\right) to get 10\left(-x^{2}\right).

$$\left(x^{2}\right)^{2}+10\left(-x^{2}\right)+16+8-3$$

Add 16 and 8 to get 24.

$$\left(x^{2}\right)^{2}+10\left(-x^{2}\right)+24-3$$

Subtract 3 from 24 to get 21.

$$\left(x^{2}\right)^{2}+10\left(-x^{2}\right)+21$$

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

$$x^{4}+10\left(-x^{2}\right)+21$$

Multiply 10 and -1 to get -10.

$$x^{4}-10x^{2}+21$$