Answer
$ \frac{x^4+3x^2+1}{x^3+2x}$.
Work Step by Step
The given expression is
$\Rightarrow x+\frac{1}{x+\frac{1}{x+\frac{1}{x}}}$
Start with the lowest denominator.
$\Rightarrow x+\frac{1}{x+\frac{1}{\frac{x}{1}+\frac{1}{x}}}$
The LCD of the lowest denominator is $x$.
Multiply the numerator and the denominator by $x$.
$\Rightarrow x+\frac{1}{x+\frac{1}{\frac{x^2}{x}+\frac{1}{x}}}$
$\Rightarrow x+\frac{1}{x+\frac{1}{\frac{x^2+1}{x}}}$
Invert the divisor and multiply.
$\Rightarrow x+\frac{1}{x+\frac{x}{x^2+1}}$
Solve the lowest denominator.
$\Rightarrow x+\frac{1}{\frac{x}{1}+\frac{x}{x^2+1}}$
The LCD of the lowest denominator is $(x^2+1)$.
Multiply the numerator and the denominator by $(x^2+1)$.
$\Rightarrow x+\frac{1}{\frac{x(x^2+1)}{(x^2+1)}+\frac{x}{x^2+1}}$
$\Rightarrow x+\frac{1}{\frac{x(x^2+1)+x}{(x^2+1)}}$
Invert the divisor and multiply.
$\Rightarrow x+\frac{(x^2+1)}{x(x^2+1)+x}$
Use the distributive property.
$\Rightarrow x+\frac{(x^2+1)}{x^3+x+x}$
Simplify.
$\Rightarrow \frac{x}{1}+\frac{(x^2+1)}{x^3+2x}$
The LCD of the denominators is $(x^3+2x)$.
Multiply the numerator and the denominator by $(x^3+2x)$.
$\Rightarrow \frac{x(x^3+2x)}{(x^3+2x)}+\frac{(x^2+1)}{(x^3+2x)}$
$\Rightarrow \frac{x(x^3+2x)+(x^2+1)}{(x^3+2x)}$
Use the distributive property.
$\Rightarrow \frac{x^4+2x^2+x^2+1}{(x^3+2x)}$
Simplify.
$\Rightarrow \frac{x^4+3x^2+1}{x^3+2x}$.
