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

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

Evaluate

$$12\left(x^{2}+1\right)$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

Combine x^{4} and -x^{4} to get 0.

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

Combine 8x^{2} and 4x^{2} to get 12x^{2}.

$$12x^{2}+16-4$$

Subtract 4 from 16 to get 12.

$$12x^{2}+12$$

Expand

$$12x^{2}+12$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

Combine x^{4} and -x^{4} to get 0.

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

Combine 8x^{2} and 4x^{2} to get 12x^{2}.

$$12x^{2}+16-4$$

Subtract 4 from 16 to get 12.

$$12x^{2}+12$$