To find the hole in the rational function \(\frac{x^2 - 25}{-3x^2 - 30x - 75}\), we need to follow these steps:

### Step 1: Factor the numerator and denominator
1. **Factor the numerator \(x^2 - 25\)**:
   \[
   x^2 - 25 = (x - 5)(x + 5)
   \]

2. **Factor the denominator \(-3x^2 - 30x - 75\)**:
   Factor out the common factor, which is \(-3\):
   \[
   -3(x^2 + 10x + 25)
   \]
   Then factor the quadratic expression:
   \[
   x^2 + 10x + 25 = (x + 5)(x + 5)
   \]
   So, the denominator becomes:
   \[
   -3(x + 5)(x + 5)
   \]

### Step 2: Simplify the expression
The function now looks like this:
\[
\frac{(x - 5)(x + 5)}{-3(x + 5)(x + 5)}
\]
We can cancel out one \((x + 5)\) term from the numerator and denominator:
\[
\frac{x - 5}{-3(x + 5)}
\]

### Step 3: Identify the hole
The hole occurs where the original function has a common factor in the numerator and denominator that is canceled out. In this case, the common factor \((x + 5)\) is canceled out, which means there is a hole at \(x = -5\).

### Step 4: Find the y-coordinate of the hole
To find the y-coordinate of the hole, substitute \(x = -5\) into the simplified function:
\[
y = \frac{-5 - 5}{-3(-5 + 5)} = \frac{-10}{-3(0)} = \text{undefined}
\]
Since substituting into the simplified expression gives an undefined result (because the denominator would be zero), this confirms the existence of the hole. The x-value of the hole is \(x = -5\).

So, the hole in the function occurs at **\(x = -5\)**.

Would you like more details, or do you have any questions? Here are some related questions:

1. How do you find the vertical asymptote in a rational function?
2. What is the difference between a hole and a vertical asymptote?
3. How do you find the horizontal asymptote of a rational function?
4. Can there be more than one hole in a rational function?
5. What happens if the numerator and denominator have no common factors?

**Tip:** Always factor both the numerator and the denominator fully before simplifying a rational expression to identify any holes or asymptotes accurately.

Learn how to find the hole in the rational function (x^2 - 25) / (-3x^2 - 30x - 75) step-by-step. This problem involves factoring the numerator and denominator, identifying where the function is undefined, and determining the x-coordinate of the hole. Suitable for high school students.

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