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

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

Evaluate

$$38+3x^{2}-x^{4}$$

Solution Steps

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

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

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

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

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

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

Use the distributive property to multiply -2 by x+2.

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

Use the distributive property to multiply -2x-4 by x-2 and combine like terms.

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

Use the distributive property to multiply -2x^{2}+8 by x^{2}+4 and combine like terms.

$$x^{4}+4x^{2}+4-2x^{4}+32-x^{2}+2$$

Combine x^{4} and -2x^{4} to get -x^{4}.

$$-x^{4}+4x^{2}+4+32-x^{2}+2$$

Add 4 and 32 to get 36.

$$-x^{4}+4x^{2}+36-x^{2}+2$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$-x^{4}+3x^{2}+36+2$$

Add 36 and 2 to get 38.

$$-x^{4}+3x^{2}+38$$

Expand

$$38+3x^{2}-x^{4}$$

Solution Steps

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

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

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

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

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

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

Use the distributive property to multiply -2 by x+2.

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

Use the distributive property to multiply -2x-4 by x-2 and combine like terms.

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

Use the distributive property to multiply -2x^{2}+8 by x^{2}+4 and combine like terms.

$$x^{4}+4x^{2}+4-2x^{4}+32-x^{2}+2$$

Combine x^{4} and -2x^{4} to get -x^{4}.

$$-x^{4}+4x^{2}+4+32-x^{2}+2$$

Add 4 and 32 to get 36.

$$-x^{4}+4x^{2}+36-x^{2}+2$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$-x^{4}+3x^{2}+36+2$$

Add 36 and 2 to get 38.

$$-x^{4}+3x^{2}+38$$