Algebra

(4x^2 - 16) / (2x^2 - 3x - 2)

Step-by-step solution with explanation

Final Answer

, where and

Step-by-step solution

1

Factor the numerator completely

First pull out the common factor of 4, then recognize as a difference of squares, which factors into .
2

Factor the denominator completely

We need two factors of the form . We need numbers that multiply to (the product of 2 and ) and add to . Those numbers are and , giving us .
3

Rewrite as one fraction with factors

Now we can clearly see the factored forms on top and bottom, which makes it easy to spot common factors.
4

Cancel the common factor

The factor appears in both the numerator and denominator, so it cancels. Note: because that would make the original denominator zero.
5

State the simplified expression and restrictions

The fully simplified form is . We exclude (canceled factor) and (makes ) because both make the original expression undefined.

Understanding this problem

Learning Insight

Canceling common factors works because any number divided by itself equals 1 — so , as long as . This is called simplifying a rational expression, and it's the same idea as reducing fractions like .

Quick Tip

Always factor FULLY before canceling. You can only cancel entire matching factors, never individual terms. If you see a difference of squares (), immediately factor it as .

Common Mistake

Students often cancel terms instead of factors — for example, incorrectly canceling the in the numerator with the in the denominator before factoring. You must factor completely first; only then can you cancel matching parenthetical groups.