Solve 2x/x+1+5/2x+3

$$\frac { 2 x } { x + 1 } + \frac { 5 } { 2 x + 3 }$$

Evaluate

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

Short Solution Steps

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

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

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

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

Do the multiplications in 2x\left(2x+3\right)+5\left(x+1\right).

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

Combine like terms in 4x^{2}+6x+5x+5.

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

Expand \left(x+1\right)\left(2x+3\right).

$$\frac{4x^{2}+11x+5}{2x^{2}+5x+3}$$

Differentiate w.r.t. x

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