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To solve this problem, we need to perform the division of the functions \(f(x)\) and \(g(x)\), and then find the domain of the resulting function.

Given:
- \( f(x) = 3x^2 + 30x + 63 \)
- \( g(x) = x + 7 \)

### Step 1: Perform the Function Division
We need to find \(\frac{f}{g}(x)\):

\[
\frac{f}{g}(x) = \frac{3x^2 + 30x + 63}{x + 7}
\]

Let's simplify this expression by factoring the numerator:

The numerator \(3x^2 + 30x + 63\) can be factored as follows:

\[
3x^2 + 30x + 63 = 3(x^2 + 10x + 21)
\]

Now, factor \(x^2 + 10x + 21\):

\[
x^2 + 10x + 21 = (x + 7)(x + 3)
\]

So, the factored form of the numerator is:

\[
3x^2 + 30x + 63 = 3(x + 7)(x + 3)
\]

Now, substitute back into the division:

\[
\frac{f}{g}(x) = \frac{3(x + 7)(x + 3)}{x + 7}
\]

Simplify by canceling out the common factor \((x + 7)\):

\[
\frac{f}{g}(x) = 3(x + 3) = 3x + 9
\]

### Step 2: Determine the Domain
The domain of the function \(\frac{f}{g}(x)\) is all real numbers except where the denominator is zero.

Since \(g(x) = x + 7\), the denominator is zero when:

\[
x + 7 = 0 \implies x = -7
\]

Therefore, the domain of \(\frac{f}{g}(x)\) is all real numbers except \(x = -7\).

**Answer:**
\[
\frac{f}{g}(x) = 3x + 9 \quad \text{with domain } x \neq -7
\]

Would you like further details, or do you have any questions?

Here are some related questions for further exploration:
1. How do you factor a quadratic expression like \(x^2 + 10x + 21\)?
2. What are the common mistakes when simplifying rational expressions?
3. How do you find the domain of a function involving division?
4. Can the function \(\frac{f}{g}(x)\) be graphed, and what does it look like?
5. How would the domain change if the denominator had more factors?

**Tip:** Always check for common factors between the numerator and denominator before simplifying rational expressions!

Let f(x) = 3x^2 + 30x + 63 and g(x) = x + 7. Perform the function operation and then find the domain of the result for f/g(x).

Learn how to perform division of rational functions with a detailed example where f(x) = 3x^2 + 30x + 63 and g(x) = x + 7. This problem covers key algebraic concepts, including simplifying rational expressions, factoring quadratic expressions, and finding the domain of functions. Suitable for high school students in Grades 9-12.

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