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To find the domain of the rational function \( F(x) = \frac{4x(x - 3)}{2x^2 - 9x - 5} \), we need to determine when the denominator is zero, as a rational function is undefined at these points.

### Step 1: Factor the Denominator
The denominator is \( 2x^2 - 9x - 5 \). Let's factor this expression to find values of \( x \) that make it zero:
\[
2x^2 - 9x - 5 = (2x + 1)(x - 5)
\]

### Step 2: Solve for the Zeroes
Now, we set each factor equal to zero to find the values of \( x \) that make the denominator zero:
1. \( 2x + 1 = 0 \Rightarrow x = -\frac{1}{2} \)
2. \( x - 5 = 0 \Rightarrow x = 5 \)

### Step 3: Determine the Domain
Since \( x = -\frac{1}{2} \) and \( x = 5 \) make the denominator zero, these values must be excluded from the domain. Therefore, the domain of \( F(x) \) is all real numbers except \( x = -\frac{1}{2} \) and \( x = 5 \).

### Answer
The correct choice is:
**A.** The domain of \( F(x) \) is \( \{ x \mid x \neq -\frac{1}{2}, 5 \} \).

Would you like further details or explanations on factoring or determining the domain?

### Related Questions
1. How do you find the domain of a general rational function?
2. What is the significance of factors in the denominator of a rational function?
3. How can we identify asymptotes from the domain restrictions?
4. What methods exist for factoring quadratic expressions?
5. How do vertical asymptotes relate to the domain of a function?

### Tip
For rational functions, always start by factoring the denominator to identify restrictions in the domain.

Find the domain of the following rational function.
F(x) = (4x(x - 3)) / (2x^2 - 9x - 5)

Learn how to find the domain of the rational function F(x) = (4x(x - 3)) / (2x^2 - 9x - 5) by factoring the denominator and identifying values that make the function undefined. This problem covers key concepts in algebra related to rational functions and domain restrictions, suitable for high school students in Grades 10-12.

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