Solve E(x)=(x/x-2-1/4-x^2+3x/x+2)*(x/4x^2-1)*frac{1}{x+2}

$$E ( x ) = ( \frac { x } { x - 2 } - \frac { 1 } { 4 - x ^ { 2 } } + \frac { 3 x } { x + 2 } ) \cdot ( \frac { x } { 4 x ^ { 2 } - 1 } ) \cdot \frac { 1 } { x + 2 }$$

Evaluate

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

Short Solution Steps

Factor 4-x^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and \left(x-2\right)\left(-x-2\right) is \left(x-2\right)\left(-x-2\right). Multiply \frac{x}{x-2} times \frac{-x-2}{-x-2}.

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

Since \frac{x\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)} and \frac{1}{\left(x-2\right)\left(-x-2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x\left(-x-2\right)-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(-x-2\right) and x+2 is \left(x-2\right)\left(x+2\right). Multiply \frac{-x^{2}-2x-1}{\left(x-2\right)\left(-x-2\right)} times \frac{-1}{-1}. Multiply \frac{3x}{x+2} times \frac{x-2}{x-2}.

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

Since \frac{-\left(-x^{2}-2x-1\right)}{\left(x-2\right)\left(x+2\right)} and \frac{3x\left(x-2\right)}{\left(x-2\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -\left(-x^{2}-2x-1\right)+3x\left(x-2\right).

$$\frac{x^{2}+2x+1+3x^{2}-6x}{\left(x-2\right)\left(x+2\right)}\times \frac{x}{4x^{2}-1}\times \frac{1}{x+2}$$

Combine like terms in x^{2}+2x+1+3x^{2}-6x.

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

Multiply \frac{4x^{2}-4x+1}{\left(x-2\right)\left(x+2\right)} times \frac{x}{4x^{2}-1} by multiplying numerator times numerator and denominator times denominator.

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

Multiply \frac{\left(4x^{2}-4x+1\right)x}{\left(x-2\right)\left(x+2\right)\left(4x^{2}-1\right)} times \frac{1}{x+2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(4x^{2}-4x+1\right)x}{\left(x-2\right)\left(x+2\right)\left(4x^{2}-1\right)\left(x+2\right)}$$

Multiply x+2 and x+2 to get \left(x+2\right)^{2}.

$$\frac{\left(4x^{2}-4x+1\right)x}{\left(x-2\right)\left(x+2\right)^{2}\left(4x^{2}-1\right)}$$

Factor the expressions that are not already factored.

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

Cancel out 2x-1 in both numerator and denominator.

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

Expand the expression.

$$\frac{2x^{2}-x}{2x^{4}+5x^{3}-6x^{2}-20x-8}$$

Expand

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

Short Solution Steps

Factor 4-x^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and \left(x-2\right)\left(-x-2\right) is \left(x-2\right)\left(-x-2\right). Multiply \frac{x}{x-2} times \frac{-x-2}{-x-2}.

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

Since \frac{x\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)} and \frac{1}{\left(x-2\right)\left(-x-2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x\left(-x-2\right)-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(-x-2\right) and x+2 is \left(x-2\right)\left(x+2\right). Multiply \frac{-x^{2}-2x-1}{\left(x-2\right)\left(-x-2\right)} times \frac{-1}{-1}. Multiply \frac{3x}{x+2} times \frac{x-2}{x-2}.

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

Since \frac{-\left(-x^{2}-2x-1\right)}{\left(x-2\right)\left(x+2\right)} and \frac{3x\left(x-2\right)}{\left(x-2\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -\left(-x^{2}-2x-1\right)+3x\left(x-2\right).

$$\frac{x^{2}+2x+1+3x^{2}-6x}{\left(x-2\right)\left(x+2\right)}\times \frac{x}{4x^{2}-1}\times \frac{1}{x+2}$$

Combine like terms in x^{2}+2x+1+3x^{2}-6x.

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

Multiply \frac{4x^{2}-4x+1}{\left(x-2\right)\left(x+2\right)} times \frac{x}{4x^{2}-1} by multiplying numerator times numerator and denominator times denominator.

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

Multiply \frac{\left(4x^{2}-4x+1\right)x}{\left(x-2\right)\left(x+2\right)\left(4x^{2}-1\right)} times \frac{1}{x+2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(4x^{2}-4x+1\right)x}{\left(x-2\right)\left(x+2\right)\left(4x^{2}-1\right)\left(x+2\right)}$$

Multiply x+2 and x+2 to get \left(x+2\right)^{2}.

$$\frac{\left(4x^{2}-4x+1\right)x}{\left(x-2\right)\left(x+2\right)^{2}\left(4x^{2}-1\right)}$$

Factor the expressions that are not already factored.

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

Cancel out 2x-1 in both numerator and denominator.

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

Expand the expression.

$$\frac{2x^{2}-x}{2x^{4}+5x^{3}-6x^{2}-20x-8}$$