Solve 4x^2+x-6/x^2+3x+2-3x/x+1+5/x+2

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

Evaluate

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

Short Solution Steps

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

$$\frac{4x^{2}+x-6}{\left(x+1\right)\left(x+2\right)}-\frac{3x}{x+1}+\frac{5}{x+2}$$

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

$$\frac{4x^{2}+x-6}{\left(x+1\right)\left(x+2\right)}-\frac{3x\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}+\frac{5}{x+2}$$

Since \frac{4x^{2}+x-6}{\left(x+1\right)\left(x+2\right)} and \frac{3x\left(x+2\right)}{\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4x^{2}+x-6-3x\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}+\frac{5}{x+2}$$

Do the multiplications in 4x^{2}+x-6-3x\left(x+2\right).

$$\frac{4x^{2}+x-6-3x^{2}-6x}{\left(x+1\right)\left(x+2\right)}+\frac{5}{x+2}$$

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

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

Factor the expressions that are not already factored in \frac{x^{2}-5x-6}{\left(x+1\right)\left(x+2\right)}.

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

Cancel out x+1 in both numerator and denominator.

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

Since \frac{x-6}{x+2} and \frac{5}{x+2} have the same denominator, add them by adding their numerators.

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

Combine like terms in x-6+5.

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

Expand

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

Short Solution Steps

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

$$\frac{4x^{2}+x-6}{\left(x+1\right)\left(x+2\right)}-\frac{3x}{x+1}+\frac{5}{x+2}$$

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

$$\frac{4x^{2}+x-6}{\left(x+1\right)\left(x+2\right)}-\frac{3x\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}+\frac{5}{x+2}$$

Since \frac{4x^{2}+x-6}{\left(x+1\right)\left(x+2\right)} and \frac{3x\left(x+2\right)}{\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4x^{2}+x-6-3x\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}+\frac{5}{x+2}$$

Do the multiplications in 4x^{2}+x-6-3x\left(x+2\right).

$$\frac{4x^{2}+x-6-3x^{2}-6x}{\left(x+1\right)\left(x+2\right)}+\frac{5}{x+2}$$

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

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

Factor the expressions that are not already factored in \frac{x^{2}-5x-6}{\left(x+1\right)\left(x+2\right)}.

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

Cancel out x+1 in both numerator and denominator.

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

Since \frac{x-6}{x+2} and \frac{5}{x+2} have the same denominator, add them by adding their numerators.

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

Combine like terms in x-6+5.

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