You may want to review reducing rational functions first.
We multiply rational functions just like fractions. The numerator of the answer is the product of the two numerators; the denominator is the product of the two denominators. For example, suppose we have to multiply −2x by (2x2 + 1) / x2. Just as for fractions, we remember that −2x is the same thing as −2x / 1. Now we multiply the numerators, getting −4x3 − 2x, and the denominators, getting x2. So the answer is (−4x3 − 2x) / x2. We should now reduce the fraction by canceling an x, so our final answer is (−4x2 − 2) / x.
In this example, I multiplied things out first. I did this to make it clearer what I was doing. In practice, though, it's probably better to leave the numerator and denominator in factored form (in this case, say, leaving the numerator as −2x(2x2 + 1)) until after you've tried to reduce the fraction. That way, you're more likely to see common factors to cancel.