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

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

Evaluate

$$\frac{2\left(2x^{3}-9x^{2}+11x+2\right)}{\left(x-3\right)\left(x+1\right)\left(x^{2}-4x+5\right)}$$

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x-3+2x+2.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+1\right) and x^{2}-4x+5 is \left(x-3\right)\left(x+1\right)\left(x^{2}-4x+5\right). Multiply \frac{3x-1}{\left(x-3\right)\left(x+1\right)} times \frac{x^{2}-4x+5}{x^{2}-4x+5}. Multiply \frac{1x-3}{x^{2}-4x+5} times \frac{\left(x-3\right)\left(x+1\right)}{\left(x-3\right)\left(x+1\right)}.

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

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

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

Do the multiplications in \left(3x-1\right)\left(x^{2}-4x+5\right)+\left(1x-3\right)\left(x-3\right)\left(x+1\right).

$$\frac{3x^{3}-12x^{2}+15x-x^{2}+4x-5+x^{3}-2x^{2}-3x-3x^{2}+6x+9}{\left(x-3\right)\left(x+1\right)\left(x^{2}-4x+5\right)}$$

Combine like terms in 3x^{3}-12x^{2}+15x-x^{2}+4x-5+x^{3}-2x^{2}-3x-3x^{2}+6x+9.

$$\frac{4x^{3}-18x^{2}+22x+4}{\left(x-3\right)\left(x+1\right)\left(x^{2}-4x+5\right)}$$

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

$$\frac{4x^{3}-18x^{2}+22x+4}{x^{4}-6x^{3}+10x^{2}+2x-15}$$

Expand

$$\frac{2\left(2x^{3}-9x^{2}+11x+2\right)}{\left(x-3\right)\left(x+1\right)\left(x^{2}-4x+5\right)}$$

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x-3+2x+2.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+1\right) and x^{2}-4x+5 is \left(x-3\right)\left(x+1\right)\left(x^{2}-4x+5\right). Multiply \frac{3x-1}{\left(x-3\right)\left(x+1\right)} times \frac{x^{2}-4x+5}{x^{2}-4x+5}. Multiply \frac{1x-3}{x^{2}-4x+5} times \frac{\left(x-3\right)\left(x+1\right)}{\left(x-3\right)\left(x+1\right)}.

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

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

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

Do the multiplications in \left(3x-1\right)\left(x^{2}-4x+5\right)+\left(1x-3\right)\left(x-3\right)\left(x+1\right).

$$\frac{3x^{3}-12x^{2}+15x-x^{2}+4x-5+x^{3}-2x^{2}-3x-3x^{2}+6x+9}{\left(x-3\right)\left(x+1\right)\left(x^{2}-4x+5\right)}$$

Combine like terms in 3x^{3}-12x^{2}+15x-x^{2}+4x-5+x^{3}-2x^{2}-3x-3x^{2}+6x+9.

$$\frac{4x^{3}-18x^{2}+22x+4}{\left(x-3\right)\left(x+1\right)\left(x^{2}-4x+5\right)}$$

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

$$\frac{4x^{3}-18x^{2}+22x+4}{x^{4}-6x^{3}+10x^{2}+2x-15}$$