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

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

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{2x^{2}+x}{x^{3}-4x}.

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

Cancel out x in both numerator and denominator.

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

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

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

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

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

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

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

Since \frac{\left(x^{2}-4x+4\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} and \frac{\left(3x-2\right)\left(x-2\right)\left(x+2\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x^{2}-4x+4\right)\left(x+1\right)-\left(3x-2\right)\left(x-2\right)\left(x+2\right).

$$\frac{x^{3}+x^{2}-4x^{2}-4x+4x+4-3x^{3}+12x+2x^{2}-8}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}$$

Combine like terms in x^{3}+x^{2}-4x^{2}-4x+4x+4-3x^{3}+12x+2x^{2}-8.

$$\frac{-2x^{3}-x^{2}+12x-4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}$$

Factor the expressions that are not already factored in \frac{-2x^{3}-x^{2}+12x-4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}.

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

Cancel out x-2 in both numerator and denominator.

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

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

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

Use the distributive property to multiply 2 by -x^{2}-\frac{5}{2}x+1.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{2x^{2}+x}{x^{3}-4x}.

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

Cancel out x in both numerator and denominator.

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

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

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

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

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

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

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

Since \frac{\left(x^{2}-4x+4\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} and \frac{\left(3x-2\right)\left(x-2\right)\left(x+2\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x^{2}-4x+4\right)\left(x+1\right)-\left(3x-2\right)\left(x-2\right)\left(x+2\right).

$$\frac{x^{3}+x^{2}-4x^{2}-4x+4x+4-3x^{3}+12x+2x^{2}-8}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}$$

Combine like terms in x^{3}+x^{2}-4x^{2}-4x+4x+4-3x^{3}+12x+2x^{2}-8.

$$\frac{-2x^{3}-x^{2}+12x-4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}$$

Factor the expressions that are not already factored in \frac{-2x^{3}-x^{2}+12x-4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}.

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

Cancel out x-2 in both numerator and denominator.

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

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

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

Use the distributive property to multiply 2 by -x^{2}-\frac{5}{2}x+1.

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