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To solve for the slope of the slant (oblique) asymptote of the rational function \( f(x) = \frac{x^3 - x^2 - 4x}{x^2 - 4x} \), follow these steps:

### Step 1: Identify the degrees of the numerator and the denominator
- The degree of the numerator, \( x^3 - x^2 - 4x \), is 3.
- The degree of the denominator, \( x^2 - 4x \), is 2.

Since the degree of the numerator (3) is greater than the degree of the denominator (2) by exactly 1, there will be a slant (oblique) asymptote.

### Step 2: Perform polynomial long division
We divide \( x^3 - x^2 - 4x \) by \( x^2 - 4x \).

1. First term: \( \frac{x^3}{x^2} = x \). Multiply \( x \) by \( x^2 - 4x \), which gives \( x^3 - 4x^2 \).
   - Subtract: \( (x^3 - x^2 - 4x) - (x^3 - 4x^2) = 3x^2 - 4x \).

2. Next term: \( \frac{3x^2}{x^2} = 3 \). Multiply \( 3 \) by \( x^2 - 4x \), which gives \( 3x^2 - 12x \).
   - Subtract: \( (3x^2 - 4x) - (3x^2 - 12x) = 8x \).

So, the quotient from the division is \( x + 3 \), and the remainder is \( 8x \).

### Step 3: Identify the slant asymptote
The equation of the slant asymptote is given by the quotient obtained in the division, which is \( y = x + 3 \). Therefore, the slope of the slant asymptote is the coefficient of \( x \), which is **1**.

### Answer:
The slope of the slant asymptote of the graph of \( f(x) \) is **1**.

Would you like further details or clarifications on any of the steps?

Here are five follow-up questions for deeper understanding:
1. How does the degree of the numerator and denominator affect the presence of a slant asymptote?
2. Why is polynomial long division necessary to find the slant asymptote?
3. How would you find the vertical asymptotes of the same function?
4. Can a rational function have both slant and horizontal asymptotes? Why or why not?
5. How would the result change if the degrees of the numerator and denominator were equal?

**Tip**: Slant asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator in a rational function.

The rational function f(x) = (x^3 - x^2 - 4x) / (x^2 - 4x). In the xy-plane, which of the following is the slope of a slant asymptote of the graph of f?

Learn how to find the slope of the slant asymptote for the rational function f(x) = (x^3 - x^2 - 4x) / (x^2 - 4x). This problem uses polynomial long division and is suitable for students in Grades 10-12.

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