Solve 1/x(x+1)+1/x+1(x+2)+1/(x+2)(x+3+1/(x+3)(x+4)=

$$\frac { 1 } { x ( x + 1 ) } + \frac { 1 } { x + 1 ( x + 2 ) } + \frac { 1 } { ( x + 2 ) ( x + 3 } + \frac { 1 } { ( x + 3 ) ( x + 4 ) } =$$

Evaluate

$$\frac{x^{3}+12x^{2}+24x+16}{2x\left(x+1\right)\left(x+2\right)\left(x+4\right)}$$

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$\frac{2x^{2}+10x+12+x^{3}+5x^{2}+6x+2x^{2}+2x}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}$$

Combine like terms in 2x^{2}+10x+12+x^{3}+5x^{2}+6x+2x^{2}+2x.

$$\frac{9x^{2}+18x+12+x^{3}}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x\left(x+1\right)\left(x+2\right)\left(x+3\right) and \left(x+3\right)\left(x+4\right) is 2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right). Multiply \frac{9x^{2}+18x+12+x^{3}}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)} times \frac{x+4}{x+4}. Multiply \frac{1}{\left(x+3\right)\left(x+4\right)} times \frac{2x\left(x+1\right)\left(x+2\right)}{2x\left(x+1\right)\left(x+2\right)}.

$$\frac{\left(9x^{2}+18x+12+x^{3}\right)\left(x+4\right)}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}+\frac{2x\left(x+1\right)\left(x+2\right)}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Since \frac{\left(9x^{2}+18x+12+x^{3}\right)\left(x+4\right)}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)} and \frac{2x\left(x+1\right)\left(x+2\right)}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(9x^{2}+18x+12+x^{3}\right)\left(x+4\right)+2x\left(x+1\right)\left(x+2\right)}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Do the multiplications in \left(9x^{2}+18x+12+x^{3}\right)\left(x+4\right)+2x\left(x+1\right)\left(x+2\right).

$$\frac{9x^{3}+36x^{2}+18x^{2}+72x+12x+48+x^{4}+4x^{3}+2x^{3}+4x^{2}+2x^{2}+4x}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Combine like terms in 9x^{3}+36x^{2}+18x^{2}+72x+12x+48+x^{4}+4x^{3}+2x^{3}+4x^{2}+2x^{2}+4x.

$$\frac{15x^{3}+60x^{2}+88x+48+x^{4}}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Factor the expressions that are not already factored in \frac{15x^{3}+60x^{2}+88x+48+x^{4}}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}.

$$\frac{\left(x+3\right)\left(x^{3}+12x^{2}+24x+16\right)}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Cancel out x+3 in both numerator and denominator.

$$\frac{x^{3}+12x^{2}+24x+16}{2x\left(x+1\right)\left(x+2\right)\left(x+4\right)}$$

Expand 2x\left(x+1\right)\left(x+2\right)\left(x+4\right).

$$\frac{x^{3}+12x^{2}+24x+16}{2x^{4}+14x^{3}+28x^{2}+16x}$$

Expand

$$\frac{x^{3}+12x^{2}+24x+16}{2x\left(x+1\right)\left(x+2\right)\left(x+4\right)}$$

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$\frac{2x^{2}+10x+12+x^{3}+5x^{2}+6x+2x^{2}+2x}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}$$

Combine like terms in 2x^{2}+10x+12+x^{3}+5x^{2}+6x+2x^{2}+2x.

$$\frac{9x^{2}+18x+12+x^{3}}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x\left(x+1\right)\left(x+2\right)\left(x+3\right) and \left(x+3\right)\left(x+4\right) is 2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right). Multiply \frac{9x^{2}+18x+12+x^{3}}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)} times \frac{x+4}{x+4}. Multiply \frac{1}{\left(x+3\right)\left(x+4\right)} times \frac{2x\left(x+1\right)\left(x+2\right)}{2x\left(x+1\right)\left(x+2\right)}.

$$\frac{\left(9x^{2}+18x+12+x^{3}\right)\left(x+4\right)}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}+\frac{2x\left(x+1\right)\left(x+2\right)}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Since \frac{\left(9x^{2}+18x+12+x^{3}\right)\left(x+4\right)}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)} and \frac{2x\left(x+1\right)\left(x+2\right)}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(9x^{2}+18x+12+x^{3}\right)\left(x+4\right)+2x\left(x+1\right)\left(x+2\right)}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Do the multiplications in \left(9x^{2}+18x+12+x^{3}\right)\left(x+4\right)+2x\left(x+1\right)\left(x+2\right).

$$\frac{9x^{3}+36x^{2}+18x^{2}+72x+12x+48+x^{4}+4x^{3}+2x^{3}+4x^{2}+2x^{2}+4x}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Combine like terms in 9x^{3}+36x^{2}+18x^{2}+72x+12x+48+x^{4}+4x^{3}+2x^{3}+4x^{2}+2x^{2}+4x.

$$\frac{15x^{3}+60x^{2}+88x+48+x^{4}}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Factor the expressions that are not already factored in \frac{15x^{3}+60x^{2}+88x+48+x^{4}}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}.

$$\frac{\left(x+3\right)\left(x^{3}+12x^{2}+24x+16\right)}{2x\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Cancel out x+3 in both numerator and denominator.

$$\frac{x^{3}+12x^{2}+24x+16}{2x\left(x+1\right)\left(x+2\right)\left(x+4\right)}$$

Expand 2x\left(x+1\right)\left(x+2\right)\left(x+4\right).

$$\frac{x^{3}+12x^{2}+24x+16}{2x^{4}+14x^{3}+28x^{2}+16x}$$