Solve frac{x^{2}+x-2}{x^{2}+5x+6}*x^2+3x/x^2-1+2x^2-4x/x^2+x-6

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

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}+x-2}{x^{2}+5x+6}.

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

Cancel out x+2 in both numerator and denominator.

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

Multiply \frac{x-1}{x+3} times \frac{x^{2}+3x}{x^{2}-1} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored in \frac{2x^{2}-4x}{x^{2}+x-6}.

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

Cancel out x-2 in both numerator and denominator.

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

Factor \left(x+3\right)\left(x^{2}-1\right).

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored in \frac{3x^{3}+2x^{2}-5x}{\left(x-1\right)\left(x+1\right)\left(x+3\right)}.

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

Cancel out x-1 in both numerator and denominator.

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

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

$$\frac{x\left(3x+5\right)}{x^{2}+4x+3}$$

Use the distributive property to multiply x by 3x+5.

$$\frac{3x^{2}+5x}{x^{2}+4x+3}$$

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}+x-2}{x^{2}+5x+6}.

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

Cancel out x+2 in both numerator and denominator.

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

Multiply \frac{x-1}{x+3} times \frac{x^{2}+3x}{x^{2}-1} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored in \frac{2x^{2}-4x}{x^{2}+x-6}.

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

Cancel out x-2 in both numerator and denominator.

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

Factor \left(x+3\right)\left(x^{2}-1\right).

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

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

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

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

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

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

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

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

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

Factor the expressions that are not already factored in \frac{3x^{3}+2x^{2}-5x}{\left(x-1\right)\left(x+1\right)\left(x+3\right)}.

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

Cancel out x-1 in both numerator and denominator.

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

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

$$\frac{x\left(3x+5\right)}{x^{2}+4x+3}$$

Use the distributive property to multiply x by 3x+5.

$$\frac{3x^{2}+5x}{x^{2}+4x+3}$$