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

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

Evaluate

$$\frac{x-1}{x+4}$$

Short Solution Steps

Factor x^{2}+5x+4.

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

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

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

Since \frac{3x^{2}+4x-5}{\left(x+1\right)\left(x+4\right)} and \frac{2x\left(x+4\right)}{\left(x+1\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3x^{2}+4x-5-2x\left(x+4\right).

$$\frac{3x^{2}+4x-5-2x^{2}-8x}{\left(x+1\right)\left(x+4\right)}+\frac{4}{x+4}$$

Combine like terms in 3x^{2}+4x-5-2x^{2}-8x.

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

Factor the expressions that are not already factored in \frac{x^{2}-4x-5}{\left(x+1\right)\left(x+4\right)}.

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

Cancel out x+1 in both numerator and denominator.

$$\frac{x-5}{x+4}+\frac{4}{x+4}$$

Since \frac{x-5}{x+4} and \frac{4}{x+4} have the same denominator, add them by adding their numerators.

$$\frac{x-5+4}{x+4}$$

Combine like terms in x-5+4.

$$\frac{x-1}{x+4}$$

Expand

$$\frac{x-1}{x+4}$$

Short Solution Steps

Factor x^{2}+5x+4.

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

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

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

Since \frac{3x^{2}+4x-5}{\left(x+1\right)\left(x+4\right)} and \frac{2x\left(x+4\right)}{\left(x+1\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3x^{2}+4x-5-2x\left(x+4\right).

$$\frac{3x^{2}+4x-5-2x^{2}-8x}{\left(x+1\right)\left(x+4\right)}+\frac{4}{x+4}$$

Combine like terms in 3x^{2}+4x-5-2x^{2}-8x.

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

Factor the expressions that are not already factored in \frac{x^{2}-4x-5}{\left(x+1\right)\left(x+4\right)}.

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

Cancel out x+1 in both numerator and denominator.

$$\frac{x-5}{x+4}+\frac{4}{x+4}$$

Since \frac{x-5}{x+4} and \frac{4}{x+4} have the same denominator, add them by adding their numerators.

$$\frac{x-5+4}{x+4}$$

Combine like terms in x-5+4.

$$\frac{x-1}{x+4}$$