Solve 2x/x+1+3/x+2+1/x^2+3x+2

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

Evaluate

$$\frac{2x^{2}+7x+4}{\left(x+1\right)\left(x+2\right)}$$

Short Solution Steps

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

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

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

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

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

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

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

$$\frac{2x^{2}+7x+3}{\left(x+1\right)\left(x+2\right)}+\frac{1}{x^{2}+3x+2}$$

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

$$\frac{2x^{2}+7x+3}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}$$

Since \frac{2x^{2}+7x+3}{\left(x+1\right)\left(x+2\right)} and \frac{1}{\left(x+1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{2x^{2}+7x+3+1}{\left(x+1\right)\left(x+2\right)}$$

Combine like terms in 2x^{2}+7x+3+1.

$$\frac{2x^{2}+7x+4}{\left(x+1\right)\left(x+2\right)}$$

Expand \left(x+1\right)\left(x+2\right).

$$\frac{2x^{2}+7x+4}{x^{2}+3x+2}$$

Differentiate w.r.t. x

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