https://r2.erweima.ai/mathbot/image/17253820894375cq6kkgf.jpg

The expression given in the image is:

\[
\frac{3x^2 - 5x - 2}{x - 2}
\]

We want to factorize the numerator and simplify the expression.

### Step 1: Factorize the Numerator
The numerator is \(3x^2 - 5x - 2\). To factorize this, we look for two numbers that multiply to \(3 \times -2 = -6\) and add up to \(-5\). The numbers are \(-6\) and \(1\).

Now, rewrite the middle term using these two numbers:

\[
3x^2 - 6x + x - 2
\]

Group the terms:

\[
3x(x - 2) + 1(x - 2)
\]

Factor out \((x - 2)\):

\[
(3x + 1)(x - 2)
\]

### Step 2: Simplify the Expression
Now, substitute the factorized form of the numerator into the expression:

\[
\frac{(3x + 1)(x - 2)}{x - 2}
\]

Simplify by canceling the \((x - 2)\) terms:

\[
3x + 1, \quad \text{for } x \neq 2
\]

Therefore, the simplified expression is \(3x + 1\) for \(x \to 2\).

Would you like more details or have any questions?

Here are some related questions you might find helpful:

1. How do you find common factors when factorizing polynomials?
2. What happens if the factor you cancel is in the denominator, and why is it important for the value \(x = 2\)?
3. How do you check if your factorization of a polynomial is correct?
4. What methods can you use to factor trinomials that don't easily factorize?
5. How would the simplification change if the numerator did not factor exactly?

**Tip:** Always check for common factors before attempting to factorize a polynomial.

Learn how to factorize the polynomial 3x^2 - 5x - 2 and simplify the rational expression (3x^2 - 5x - 2) / (x - 2). This problem involves polynomial factorization and simplification techniques. Suitable for high school students.

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