https://file.aiquickdraw.com/mathbot/file/1728885631687o9zkojvj.jpg

The expression shown in the image is:

\[
\frac{(5x^2 - 17x - 15)}{(x - 4)}
\]

To simplify this expression, we will factor the quadratic expression in the numerator and then perform the division.

### Step 1: Factor the quadratic expression \( 5x^2 - 17x - 15 \).

We need two numbers that multiply to \( 5 \times (-15) = -75 \) and add to \( -17 \). These numbers are \( -25 \) and \( 3 \), because \( -25 \times 3 = -75 \) and \( -25 + 3 = -17 \).

We can rewrite the middle term:

\[
5x^2 - 25x + 3x - 15
\]

Now, group the terms and factor:

\[
5x(x - 5) + 3(x - 5)
\]

Factor out \( (x - 5) \):

\[
(5x + 3)(x - 5)
\]

### Step 2: Rewrite the entire expression.

Now the expression becomes:

\[
\frac{(5x + 3)(x - 5)}{(x - 4)}
\]

This is the simplified form of the expression. Since \( x - 4 \) does not cancel with anything in the numerator, this is the final result.

Would you like to explore this in more detail or ask any other questions?

### Related Questions:
1. How do you factor quadratic expressions in general?
2. What is the process for solving a rational expression like this when there are common factors?
3. What happens if the denominator had common factors with the numerator?
4. How can this expression be used to find the vertical asymptotes of a rational function?
5. What restrictions should be placed on the variable \( x \) in this expression?

### Tip:
When factoring a quadratic, always check if the factors can simplify the rational expression by canceling common terms.

Simplify the expression (5x^2 - 17x - 15) ÷ (x - 4).

Learn how to simplify the rational expression (5x^2 - 17x - 15) ÷ (x - 4) by factoring the quadratic expression. This solution walks through the step-by-step process, including factoring and simplifying. Suitable for high school students in Grades 9-11.

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