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

, where $x eq -2, -3, 2$

Step-by-step solution

Factor every polynomial completely

We factor the numerators and denominators so we can cancel common factors. Recognizing 3x²-12 as a difference of squares (after pulling out 3) and x²+5x+6 as a product of two binomials makes this possible.

Rewrite multiplication with factored forms

We substitute the factored forms into the first product. This sets us up to cancel matching factors in the numerator and denominator.

Cancel common factors in the product

The 3's cancel, one (x+2) cancels, and one (x-2) cancels from (x-2)². We're left with a much simpler fraction.

Rewrite the full expression with common denominator

Both fractions already share the denominator (x-2), so we can add them directly without any extra work.

Expand the numerator of the first fraction

Multiply the two binomials using FOIL: x·x + x·3 + 2·x + 2·3 = x²+3x+2x+6 = x²+5x+6.

Add the numerators over the common denominator

Since the denominators match, we just add the numerators. Combining the constant terms: 6+10 = 16.

Check if numerator factors further (it does not)

We check the discriminant to see if x²+5x+16 factors over real numbers. Since it's negative, the numerator cannot be factored, so this is the fully simplified answer.

Understanding this problem

Learning Insight

Multiplying rational expressions is really about canceling shared factors — just like simplifying regular fractions. The key insight is that factoring first reveals 'hidden' cancellations that make the problem much simpler before you even multiply.

Quick Tip

Always factor BEFORE you multiply or add fractions. Trying to cancel after expanding everything is much harder and leads to errors. Factor → Cancel → Then combine.

Common Mistake

Students often try to cancel terms instead of factors — for example, canceling the x² in the numerator with an x² in the denominator when they are part of a sum like x²+5x+6, not a standalone factor. You can only cancel factors that are multiplied, never terms that are added or subtracted.