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To solve the problem, we need to find:

1. The domain of the function \( f(x) = \frac{4x^2 - 4x - 1}{2x - 3} \).
2. The equation of the slant asymptote.

### Step 1: Domain in interval notation
The domain of a rational function is determined by the values that make the denominator equal to zero, as division by zero is undefined. The denominator of the function is \( 2x - 3 \). Set it equal to zero:

\[
2x - 3 = 0 \implies x = \frac{3}{2}
\]

Thus, \( x = \frac{3}{2} \) is not in the domain. The domain excludes this value and is expressed in interval notation as:

\[
(-\infty, \frac{3}{2}) \cup (\frac{3}{2}, \infty)
\]

### Step 2: Slant asymptote
A slant asymptote exists when the degree of the numerator is exactly one higher than the degree of the denominator. In this case, the numerator is quadratic (\(4x^2 - 4x - 1\)) and the denominator is linear (\(2x - 3\)).

To find the equation of the slant asymptote, we perform polynomial long division of \(4x^2 - 4x - 1\) by \(2x - 3\).

#### Division process:
1. Divide the leading term of the numerator (\(4x^2\)) by the leading term of the denominator (\(2x\)) to get \(2x\).
2. Multiply \(2x\) by \(2x - 3\) to get \(4x^2 - 6x\).
3. Subtract \(4x^2 - 6x\) from the original numerator \(4x^2 - 4x - 1\), resulting in \(2x - 1\).
4. Divide \(2x\) by \(2x\) to get \(1\).
5. Multiply \(1\) by \(2x - 3\) to get \(2x - 3\).
6. Subtract \(2x - 3\) from \(2x - 1\), resulting in \(2\).

So the quotient is \(2x + 1\) and the remainder is 2. The equation of the slant asymptote is given by the quotient, which is:

\[
y = 2x + 1
\]

### Final answers:
- The domain in interval notation is: \[(-\infty, \frac{3}{2}) \cup (\frac{3}{2}, \infty)\]
- The equation of the slant asymptote is: \[y = 2x + 1\]

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Would you like further clarification on any of these steps?

Here are five related questions you might explore:
1. How do you find vertical asymptotes for rational functions?
2. What is the difference between a horizontal and slant asymptote?
3. How can we determine whether a rational function has a hole?
4. What happens

Find the equation of the domain in interval notation and the equation of the slant asymptote of the rational function: f(x) = (4x^2 - 4x - 1) / (2x - 3).

Learn how to find the domain and slant asymptote of the rational function f(x) = (4x^2 - 4x - 1) / (2x - 3) with detailed steps, including determining the domain by excluding values that make the denominator zero and finding the slant asymptote using polynomial long division. Suitable for Grades 10-12.

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