Multiplying and dividing rational expressions follows the rules for fractions: multiply straight across, and divide by multiplying by the reciprocal. The new habit from Simplifying Rational Expressions is to factor everything first, so you can cancel before anything gets big.
Key idea
Factor every numerator and denominator. To divide, flip the second fraction and multiply. Cancel common factors anywhere between a numerator and a denominator, then multiply what is left. Excluded values include every $x$ that made any denominator 0, including the numerator of a fraction you flipped.
Why it works
$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$ is how fractions multiply, and dividing by $\frac{c}{d}$ means multiplying by $\frac{d}{c}$, because $\frac{c}{d} \cdot \frac{d}{c} = 1$. Cancelling before multiplying is allowed because multiplication can be done in any order. When you flip $\frac{c}{d}$, its numerator becomes a denominator, so values making $c = 0$ are excluded too.
Worked examples
Worked example: Multiplying
Simplify $\frac{x^2 + x - 2}{x^2 - x - 12} \cdot \frac{x + 3}{x - 1}$.
- Factor: $\frac{(x - 1)(x + 2)}{(x + 3)(x - 4)} \cdot \frac{x + 3}{x - 1}$.
- Cancel $(x - 1)$ and $(x + 3)$.
Answer: $\frac{x + 2}{x - 4}$, for $x \ne 1, -3, 4$
Worked example: Dividing
Simplify $\frac{x^2 - 4}{3x} \div \frac{x + 2}{6x^2}$.
- Flip and multiply: $\frac{(x - 2)(x + 2)}{3x} \cdot \frac{6x^2}{x + 2}$.
- Cancel $(x + 2)$, and $\frac{6x^2}{3x} = 2x$.
Answer: $2x(x - 2)$, for $x \ne 0, -2$
Common mistake
Cancelling before you factor, for example crossing out the $x^2$ in $\frac{x^2 - 4}{x^2 + 2x}$. Terms inside a sum can't be cancelled. Factor first: $\frac{(x - 2)(x + 2)}{x(x + 2)} = \frac{x - 2}{x}$. Check with $x = 1$: $\frac{-3}{3} = -1$ and $\frac{-1}{1} = -1$.