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We are given the expression:

\[
\frac{15x^2 - 40x}{5x}
\]

and are asked to find the quotient for all values of \( x \) where the expression is defined. Let's simplify the expression step by step.

### Step 1: Factor out the greatest common factor (GCF)
First, notice that we can factor the numerator. The GCF of \( 15x^2 \) and \( -40x \) is \( 5x \), so we factor it out:

\[
15x^2 - 40x = 5x(3x - 8)
\]

Now the expression becomes:

\[
\frac{5x(3x - 8)}{5x}
\]

### Step 2: Simplify the expression
We can now cancel out the \( 5x \) from the numerator and the denominator (as long as \( x \neq 0 \), because division by zero is undefined):

\[
\frac{5x(3x - 8)}{5x} = 3x - 8
\]

Thus, the quotient is \( 3x - 8 \), which corresponds to **Option C**.

### Final Answer: C. \( 3x - 8 \)

Would you like further clarification on this process? Here are some related questions you can consider:

1. What happens if \( x = 0 \) in this expression?
2. How do you find the domain of a rational expression?
3. Why is it important to factor before simplifying?
4. Can this expression be factored differently?
5. How do you handle expressions with multiple variables in the numerator?

**Tip:** Always check for restrictions in the denominator when simplifying rational expressions to avoid division by zero errors.

What is the quotient represented by the expression (15x^2 - 40x) / (5x) for all values of x where the expression is defined?

This page provides a step-by-step solution to simplify the rational expression (15x^2 - 40x) / (5x). Learn how factoring and simplifying rational expressions are used to find the quotient. Suitable for students in Grades 9-10.

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