Solve x+1div(x+3x^2+3x+1)

$$x+1 \div (x+3 { x }^{ 2 } +3x+1)$$

Evaluate

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

Short Solution Steps

Combine x and 3x to get 4x.

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

Factor 4x+3x^{2}+1.

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

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

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

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

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

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

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

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

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

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

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

Differentiate w.r.t. x

$$\frac{9x^{4}+24x^{3}+22x^{2}+2x-3}{\left(\left(x+1\right)\left(3x+1\right)\right)^{2}}$$