How do I rewrite this rational expression? How do I rewrite the rational expression: 
$$\frac{x^3+5x^2+3x-10}{x+4}$$
But in the form of: 
$$q(x) + \frac{r(x)}{b(x)}$$

Use Polynomial long division. The only way I know how to format that here would be something like:
$$\begin{split}
x^3 + 5x^2 + 3x - 10 &= (x+4)(x^2) + x^2 + 3x - 10 \\
&= (x+4)(x^2) + (x+4)(x) - x - 10 \\
&= (x+4)(x^2) + (x+4)(x) - (x+4)(1) - 6 \\
&= (x+4)(x^2 + x - 1) - 6
\end{split}$$
So:
$$\frac{x^3 + 5x^2 + 3x - 10}{x+4} = (x^2 + x - 1) + \frac{-6}{x+4}$$
