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We are given the rational function \( f(x) = \frac{x^2 - 3x - 28}{x^2 + x - 30} \), and the task is to find its completely factored form.

### Step 1: Factor the numerator \( x^2 - 3x - 28 \).
We are looking for two numbers that multiply to \(-28\) and add up to \(-3\).

- These numbers are \(-7\) and \(4\), since \( (-7)(4) = -28 \) and \( (-7) + 4 = -3 \).

Thus, the numerator factors as:
\[
x^2 - 3x - 28 = (x - 7)(x + 4)
\]

### Step 2: Factor the denominator \( x^2 + x - 30 \).
We need two numbers that multiply to \(-30\) and add up to \(1\).

- These numbers are \(6\) and \(-5\), since \( 6(-5) = -30 \) and \( 6 + (-5) = 1 \).

Thus, the denominator factors as:
\[
x^2 + x - 30 = (x + 6)(x - 5)
\]

### Step 3: Combine the factored numerator and denominator.
The completely factored form of the rational function is:
\[
f(x) = \frac{(x - 7)(x + 4)}{(x + 6)(x - 5)}
\]

### Step 4: Match with the options.
Comparing this with the given options, the correct answer is:

\[
\boxed{D. \ f(x) = \frac{(x+4)(x-7)}{(x-5)(x+6)}}
\]

Would you like further explanation or more details?

Here are five related questions:
1. How do you factor a quadratic expression like \(x^2 - 3x - 28\)?
2. What are the steps to factor the denominator of a rational function?
3. How do you check if a factored form is correct?
4. What does it mean for a rational function to be completely factored?
5. How can you simplify a rational function after factoring?

**Tip:** Always verify your factors by multiplying them back to ensure they give the original quadratic expression.

What is the completely factored form of the rational function f(x) = (x^2 - 3x - 28) / (x^2 + x - 30)?

Learn how to find the completely factored form of the rational function f(x) = (x^2 - 3x - 28) / (x^2 + x - 30). This problem covers factoring quadratics and simplifying rational functions, suitable for students in Grades 9-11.

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