Find the domain of the following rational function.
Upper F left parenthesis x right parenthesis equals StartFraction 8 x left parenthesis x minus 4 right parenthesis Over 6 x squared minus 41 x minus 7 EndFraction

We are tasked with finding the domain of the rational function:

\[
F(x) = \frac{8x(x - 4)}{6x^2 - 41x - 7}
\]

### Steps:

1. **Identify restrictions on the domain:**
   The domain of a rational function consists of all real numbers **except** where the denominator equals zero (because division by zero is undefined).

   The denominator is:
   \[
   6x^2 - 41x - 7
   \]

   To find the values of \( x \) that are excluded from the domain, we set the denominator equal to zero and solve for \( x \):

   \[
   6x^2 - 41x - 7 = 0
   \]

2. **Solve the quadratic equation:**
   We can use the quadratic formula to solve this equation. The quadratic formula is:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   For the equation \( 6x^2 - 41x - 7 = 0 \), the coefficients are:
   - \( a = 6 \)
   - \( b = -41 \)
   - \( c = -7 \)

   Plugging these values into the quadratic formula:

   \[
   x = \frac{-(-41) \pm \sqrt{(-41)^2 - 4(6)(-7)}}{2(6)}
   \]
   \[
   x = \frac{41 \pm \sqrt{1681 + 168}}{12}
   \]
   \[
   x = \frac{41 \pm \sqrt{1849}}{12}
   \]
   \[
   x = \frac{41 \pm 43}{12}
   \]

3. **Find the two solutions:**
   - \( x_1 = \frac{41 + 43}{12} = \frac{84}{12} = 7 \)
   - \( x_2 = \frac{41 - 43}{12} = \frac{-2}{12} = -\frac{1}{6} \)

   Therefore, \( x = 7 \) and \( x = -\frac{1}{6} \) are the values that make the denominator zero.

4. **Conclusion:**
   The domain of \( F(x) \) consists of all real numbers **except** \( x = 7 \) and \( x = -\frac{1}{6} \).

   Hence, the domain is:
   \[
   \boxed{(-\infty, -\frac{1}{6}) \cup (-\frac{1}{6}, 7) \cup (7, \infty)}
   \]

Would you like further details or clarifications? 

Here are some related questions to practice:

1. What is the domain of \( f(x) = \frac{x^2 - 4}{x^2 - 9} \)?
2. Find the domain of \( f(x) = \frac{1}{x(x-3)} \).
3. How do you find the domain of a radical function like \( g(x) = \sqrt{x - 2} \)?
4. What happens if both the numerator and denominator of a rational function share a common factor?
5. Can the range of a rational function be restricted by the domain?

**Tip:** Always check the denominator of a rational function when finding the domain to ensure you're not dividing by zero!

Find the domain of the following rational function: F(x) = (8x(x - 4)) / (6x^2 - 41x - 7)

Find the domain of the rational function F(x) = (8x(x-4)) / (6x^2 - 41x - 7). This step-by-step guide walks through solving for values that make the denominator zero, using the quadratic formula. Ideal for students in Grades 10-12.

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