Solve 4/4+x+x/1+x

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

Evaluate

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

Short Solution Steps

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

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

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

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

Do the multiplications in 4\left(x+1\right)+x\left(x+4\right).

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

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

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

Expand \left(x+1\right)\left(x+4\right).

$$\frac{8x+4+x^{2}}{x^{2}+5x+4}$$

Differentiate w.r.t. x

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