Algebra

(3x^2 - 12)/(x + 2) · (x^2 + 5x + 6)/(3(x - 2)^2) ÷ 10/(x - 2)

Step-by-step solution with explanation

Final Answer

Step-by-step solution

Rewrite division as multiplication by reciprocal

When you divide by a fraction, flip it upside down and multiply instead. So dividing by becomes multiplying by . This makes the problem easier to work with.

Factor all numerators and denominators completely

We factor out 3 from to get . Then we recognize is a difference of squares: . We also factor into by finding two numbers that multiply to 6 and add to 5.

Expand the factored difference of squares

We write as so we can see all the factors clearly. This helps us spot what cancels out in the next steps.

Cancel common factors from numerator and denominator

When the same factor appears on top and bottom, they cancel to 1. We cancel with , appears three times on top and twice on bottom so one remains on top, and appears twice on top and twice on bottom so they all cancel.

Write the simplified expression

After canceling, we have one and one on top, and one and 10 on the bottom. But wait—we need to recount our cancellations more carefully.

Recount factors after cancellation carefully

Let's track this correctly: the three factors on top cancel with two factors on bottom, leaving none. The two on top cancel with two of the on bottom (from and the third fraction), leaving one on bottom. We're left with just on top and on bottom.

Understanding this problem

Learning Insight

When multiplying and dividing fractions, you can cancel ANY factor that appears in both a numerator and a denominator—even if they're in different fractions. Factoring completely first lets you see all possible cancellations. Think of it like reducing to by canceling the common factor of 2, but with algebraic expressions instead of numbers.

Quick Tip

Before doing any multiplication, factor everything you can and change all division to multiplication. Then cancel before multiplying—this saves you from dealing with huge expressions and makes mistakes less likely.

Common Mistake

Students often forget to change division into multiplication by the reciprocal in the very first step, or they try to cancel terms that are added or subtracted (like canceling from , which is wrong). You can only cancel factors that are multiplied, never terms that are added.