Solve =frac{(x^{2}+2)^{3}(2x)-x^{2}(3)(x^{2}+2)^2(2x)}{[(x^2+2)^1]^2}

$$= \frac { ( x ^ { 2 } + 2 ) ^ { 3 } ( 2 x ) - x ^ { 2 } ( 3 ) ( x ^ { 2 } + 2 ) ^ { 2 } ( 2 x ) } { [ ( x ^ { 2 } + 2 ) ^ { 1 } ] ^ { 2 } }$$

Evaluate

$$4x-4x^{3}$$

Short Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

To raise a power to another power, multiply the exponents. Multiply 1 and 2 to get 2.

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

Multiply 3 and 2 to get 6.

$$\frac{\left(x^{2}+2\right)^{3}\times 2x-x^{3}\times 6\left(x^{2}+2\right)^{2}}{\left(x^{2}+2\right)^{2}}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x^{2}+2\right)^{2} in both numerator and denominator.

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

Expand the expression.

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

Expand

$$4x-4x^{3}$$

Short Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

To raise a power to another power, multiply the exponents. Multiply 1 and 2 to get 2.

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

Multiply 3 and 2 to get 6.

$$\frac{\left(x^{2}+2\right)^{3}\times 2x-x^{3}\times 6\left(x^{2}+2\right)^{2}}{\left(x^{2}+2\right)^{2}}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x^{2}+2\right)^{2} in both numerator and denominator.

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

Expand the expression.

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