To simplify the function 

\[
f(x) = \frac{2x^2 - 5x - 3}{x^2 - 8x + 15},
\]

we can attempt to factor both the numerator and the denominator.

### Step 1: Factor the numerator
The numerator is \( 2x^2 - 5x - 3 \). We need to find two numbers that multiply to \( 2 \times (-3) = -6 \) and add up to \( -5 \).

The factors of \( -6 \) that add up to \( -5 \) are \( -6 \) and \( 1 \).

So, we can rewrite the middle term and factor by grouping:

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

Now, factor by grouping:

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

### Step 2: Factor the denominator
The denominator is \( x^2 - 8x + 15 \). We need to find two numbers that multiply to \( 15 \) and add up to \( -8 \).

The factors of \( 15 \) that add up to \( -8 \) are \( -5 \) and \( -3 \).

So, we can factor the denominator as:

\[
x^2 - 8x + 15 = (x - 5)(x - 3)
\]

### Step 3: Simplify the function
Now that we have factored both the numerator and the denominator, the function becomes:

\[
f(x) = \frac{(2x + 1)(x - 3)}{(x - 5)(x - 3)}
\]

Since \( (x - 3) \) appears in both the numerator and the denominator, it can be canceled out (as long as \( x \neq 3 \), where the function would be undefined).

So, the simplified function is:

\[
f(x) = \frac{2x + 1}{x - 5}, \quad \text{for } x \neq 3
\]

This is the simplified form of the given function.

Would you like to know more about the properties of this simplified function or have any other questions?

### Related Questions:
1. What are the domain and range of the simplified function?
2. How do you find the vertical asymptote of the simplified function?
3. What are the intercepts of the simplified function?
4. How can you graph the simplified function?
5. How does the simplified function differ from the original in terms of behavior and undefined points?

**Tip:** Always be cautious when canceling terms in rational functions, as it may introduce or remove certain restrictions on the domain.

Learn how to simplify the rational function f(x) = (2x^2 - 5x - 3) / (x^2 - 8x + 15) using factorization. This problem covers rational functions, factoring techniques, and function simplification methods. Suitable for high school students and those studying algebra.

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