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To find the domain of the given rational function:

\[
p(x) = \frac{-7 - 2x}{5x^2 - 31x - 72}
\]

we need to identify when the denominator is equal to zero because division by zero is undefined. So, we solve:

\[
5x^2 - 31x - 72 = 0
\]

### Step 1: Solve the quadratic equation.
We can solve this quadratic using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = 5\), \(b = -31\), and \(c = -72\).

\[
x = \frac{-(-31) \pm \sqrt{(-31)^2 - 4(5)(-72)}}{2(5)}
\]
\[
x = \frac{31 \pm \sqrt{961 + 1440}}{10}
\]
\[
x = \frac{31 \pm \sqrt{2401}}{10}
\]
\[
x = \frac{31 \pm 49}{10}
\]

### Step 2: Find the two solutions.

\[
x_1 = \frac{31 + 49}{10} = \frac{80}{10} = 8
\]
\[
x_2 = \frac{31 - 49}{10} = \frac{-18}{10} = -1.8
\]

### Step 3: Domain of the function.
The domain consists of all real numbers except where the denominator equals zero. So, the function is undefined at \(x = 8\) and \(x = -1.8\).

Therefore, the domain in interval notation is:

\[
(-\infty, -1.8) \cup (-1.8, 8) \cup (8, \infty)
\]

Would you like further details or clarification?

Here are 5 related questions:
1. What happens to the function behavior near \(x = 8\) and \(x = -1.8\)?
2. How do vertical asymptotes relate to the domain restrictions in rational functions?
3. Can you explain why dividing by zero is undefined?
4. How would the domain change if there were additional factors in the denominator?
5. What is the graphical representation of domain restrictions?

**Tip**: Always check for domain restrictions by examining the denominator of a rational function; it must not be zero!

Find the domain of the given function: p(x) = (-7 - 2x) / (5x^2 - 31x - 72).

Learn how to find the domain of the rational function (-7 - 2x) / (5x^2 - 31x - 72) by solving the quadratic equation in the denominator. This problem involves key algebraic concepts, including rational functions and the quadratic formula. Suitable for students in Grades 10-12.

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