How do I rewrite this rational expression?

Question

How do I rewrite the rational expression:

$$\frac{x^3+5x^2+3x-10}{x+4}$$

But in the form of:

$$q(x) + \frac{r(x)}{b(x)}$$

Answer 1

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:

$$\frac{x^3 + 5x^2 + 3x - 10}{x+4} = (x^2 + x - 1) + \frac{-6}{x+4}$$