Solve x/x+2-4x+2-4x/2-2x

$$\frac { x } { x + 2 - 4 x } + \frac { 2 - 4 x } { 2 - 2 x }$$

Evaluate

$$\frac{5x^{2}-6x+2}{\left(1-x\right)\left(2-3x\right)}$$

Short Solution Steps

Combine x and -4x to get -3x.

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

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

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

Cancel out 2 in both numerator and denominator.

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

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

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

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

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

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

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

Combine like terms in -x^{2}+x+6x^{2}-4x-3x+2.

$$\frac{5x^{2}-6x+2}{\left(-3x+2\right)\left(-x+1\right)}$$

Expand \left(-3x+2\right)\left(-x+1\right).

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

Expand

$$\frac{-5x^{2}+6x-2}{\left(2-3x\right)\left(x-1\right)}$$

Short Solution Steps

Combine x and -4x to get -3x.

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

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

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

Cancel out 2 in both numerator and denominator.

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

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

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

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

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

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

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

Combine like terms in -x^{2}+x+6x^{2}-4x-3x+2.

$$\frac{5x^{2}-6x+2}{\left(-3x+2\right)\left(-x+1\right)}$$

Expand \left(-3x+2\right)\left(-x+1\right).

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