Solve f(x)=1+1/x+1+1/(x+1)^2

$$f ( x ) = 1 + \frac { 1 } { x + 1 } + \frac { 1 } { ( x + 1 ) ^ { 2 } }$$

Evaluate

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

Short Solution Steps

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

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

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

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

Combine like terms in x+1+1.

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

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

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

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

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

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

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

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

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

Expand \left(x+1\right)^{2}.

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

Expand

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

Short Solution Steps

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

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

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

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

Combine like terms in x+1+1.

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

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

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

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

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

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

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

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

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

Expand \left(x+1\right)^{2}.

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