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

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

Evaluate

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

Short Solution Steps

Factor x^{2}+4x+3.

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

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

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

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

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

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

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

Combine like terms in 2x+6-1.

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

Factor x^{2}+x.

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

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

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

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

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

Combine like terms in x+1-x.

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

Divide \frac{2x+5}{\left(x+1\right)\left(x+3\right)} by \frac{1}{\left(x+1\right)x^{2}} by multiplying \frac{2x+5}{\left(x+1\right)\left(x+3\right)} by the reciprocal of \frac{1}{\left(x+1\right)x^{2}}.

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

Cancel out x+1 in both numerator and denominator.

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

Use the distributive property to multiply 2x+5 by x^{2}.

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

Expand

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

Short Solution Steps

Factor x^{2}+4x+3.

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

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

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

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

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

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

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

Combine like terms in 2x+6-1.

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

Factor x^{2}+x.

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

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

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

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

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

Combine like terms in x+1-x.

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

Divide \frac{2x+5}{\left(x+1\right)\left(x+3\right)} by \frac{1}{\left(x+1\right)x^{2}} by multiplying \frac{2x+5}{\left(x+1\right)\left(x+3\right)} by the reciprocal of \frac{1}{\left(x+1\right)x^{2}}.

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

Cancel out x+1 in both numerator and denominator.

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

Use the distributive property to multiply 2x+5 by x^{2}.

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