Solve frac{(x^{2}-1)^{4}(2x)-x^{2}(4)(x^2-1)^3(2x)}{(x^2-1)^8}

$$\frac { ( x ^ { 2 } - 1 ) ^ { 4 } ( 2 x ) - x ^ { 2 } ( 4 ) ( x ^ { 2 } - 1 ) ^ { 3 } ( 2 x ) } { ( x ^ { 2 } - 1 ) ^ { 8 } }$$

Evaluate

$$-\frac{2x\left(3x^{2}+1\right)}{\left(x^{2}-1\right)^{5}}$$

Short Solution Steps

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

$$\frac{\left(x^{2}-1\right)^{4}\times 2x-x^{3}\times 4\left(x^{2}-1\right)^{3}\times 2}{\left(x^{2}-1\right)^{8}}$$

Multiply 4 and 2 to get 8.

$$\frac{\left(x^{2}-1\right)^{4}\times 2x-x^{3}\times 8\left(x^{2}-1\right)^{3}}{\left(x^{2}-1\right)^{8}}$$

Factor the expressions that are not already factored.

$$\frac{2x\left(-3x^{2}-1\right)\left(x^{2}-1\right)^{3}}{\left(x^{2}-1\right)^{8}}$$

Cancel out \left(x^{2}-1\right)^{3} in both numerator and denominator.

$$\frac{2x\left(-3x^{2}-1\right)}{\left(x^{2}-1\right)^{5}}$$

Expand the expression.

$$\frac{-6x^{3}-2x}{x^{10}-5x^{8}+10x^{6}-10x^{4}+5x^{2}-1}$$

Expand

$$-\frac{2\left(3x^{3}+x\right)}{\left(x^{2}-1\right)^{5}}$$

Short Solution Steps

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

$$\frac{\left(x^{2}-1\right)^{4}\times 2x-x^{3}\times 4\left(x^{2}-1\right)^{3}\times 2}{\left(x^{2}-1\right)^{8}}$$

Multiply 4 and 2 to get 8.

$$\frac{\left(x^{2}-1\right)^{4}\times 2x-x^{3}\times 8\left(x^{2}-1\right)^{3}}{\left(x^{2}-1\right)^{8}}$$

Factor the expressions that are not already factored.

$$\frac{2x\left(-3x^{2}-1\right)\left(x^{2}-1\right)^{3}}{\left(x^{2}-1\right)^{8}}$$

Cancel out \left(x^{2}-1\right)^{3} in both numerator and denominator.

$$\frac{2x\left(-3x^{2}-1\right)}{\left(x^{2}-1\right)^{5}}$$

Expand the expression.

$$\frac{-6x^{3}-2x}{x^{10}-5x^{8}+10x^{6}-10x^{4}+5x^{2}-1}$$