Solve 1-2x/4x^2-1-x+4/2x^2-3x+1+1/1-x

$$\frac { 1 - 2 x } { 4 x ^ { 2 } - 1 } - \frac { x + 4 } { 2 x ^ { 2 } - 3 x + 1 } + \frac { 1 } { 1 - x }$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{1-2x}{4x^{2}-1}.

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

Extract the negative sign in 1-2x.

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

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

$$\frac{-1}{2x+1}-\frac{x+4}{2x^{2}-3x+1}+\frac{1}{1-x}$$

Factor 2x^{2}-3x+1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x+1 and \left(x-1\right)\left(2x-1\right) is \left(x-1\right)\left(2x-1\right)\left(2x+1\right). Multiply \frac{-1}{2x+1} times \frac{\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}. Multiply \frac{x+4}{\left(x-1\right)\left(2x-1\right)} times \frac{2x+1}{2x+1}.

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

Since \frac{-\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)} and \frac{\left(x+4\right)\left(2x+1\right)}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -\left(x-1\right)\left(2x-1\right)-\left(x+4\right)\left(2x+1\right).

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

Combine like terms in -2x^{2}+x+2x-1-2x^{2}-x-8x-4.

$$\frac{-4x^{2}-6x-5}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)}+\frac{1}{1-x}$$

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

$$\frac{-4x^{2}-6x-5}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)}+\frac{-\left(2x-1\right)\left(2x+1\right)}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)}$$

Since \frac{-4x^{2}-6x-5}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)} and \frac{-\left(2x-1\right)\left(2x+1\right)}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{-4x^{2}-6x-5-\left(2x-1\right)\left(2x+1\right)}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)}$$

Do the multiplications in -4x^{2}-6x-5-\left(2x-1\right)\left(2x+1\right).

$$\frac{-4x^{2}-6x-5-4x^{2}-2x+2x+1}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)}$$

Combine like terms in -4x^{2}-6x-5-4x^{2}-2x+2x+1.

$$\frac{-8x^{2}-6x-4}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)}$$

Expand \left(x-1\right)\left(2x-1\right)\left(2x+1\right).

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{1-2x}{4x^{2}-1}.

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

Extract the negative sign in 1-2x.

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

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

$$\frac{-1}{2x+1}-\frac{x+4}{2x^{2}-3x+1}+\frac{1}{1-x}$$

Factor 2x^{2}-3x+1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x+1 and \left(x-1\right)\left(2x-1\right) is \left(x-1\right)\left(2x-1\right)\left(2x+1\right). Multiply \frac{-1}{2x+1} times \frac{\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}. Multiply \frac{x+4}{\left(x-1\right)\left(2x-1\right)} times \frac{2x+1}{2x+1}.

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

Since \frac{-\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)} and \frac{\left(x+4\right)\left(2x+1\right)}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -\left(x-1\right)\left(2x-1\right)-\left(x+4\right)\left(2x+1\right).

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

Combine like terms in -2x^{2}+x+2x-1-2x^{2}-x-8x-4.

$$\frac{-4x^{2}-6x-5}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)}+\frac{1}{1-x}$$

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

$$\frac{-4x^{2}-6x-5}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)}+\frac{-\left(2x-1\right)\left(2x+1\right)}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)}$$

Since \frac{-4x^{2}-6x-5}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)} and \frac{-\left(2x-1\right)\left(2x+1\right)}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{-4x^{2}-6x-5-\left(2x-1\right)\left(2x+1\right)}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)}$$

Do the multiplications in -4x^{2}-6x-5-\left(2x-1\right)\left(2x+1\right).

$$\frac{-4x^{2}-6x-5-4x^{2}-2x+2x+1}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)}$$

Combine like terms in -4x^{2}-6x-5-4x^{2}-2x+2x+1.

$$\frac{-8x^{2}-6x-4}{\left(x-1\right)\left(2x-1\right)\left(2x+1\right)}$$

Expand \left(x-1\right)\left(2x-1\right)\left(2x+1\right).

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