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To solve the expression \(\frac{f(x)}{k(x)}\), let's begin by identifying the functions:

- \(f(x) = x + 2\)
- \(k(x) = x^2 - 6x - 16\)

### Step 1: Setting up the division
The problem asks for:
\[
\frac{f(x)}{k(x)} = \frac{x + 2}{x^2 - 6x - 16}
\]

### Step 2: Factor the denominator if possible
The quadratic expression in the denominator can be factored. We need to factor \(x^2 - 6x - 16\).

To factor, we look for two numbers that multiply to \(-16\) and add to \(-6\). These numbers are \(-8\) and \(2\).

Thus, the factorization is:
\[
x^2 - 6x - 16 = (x - 8)(x + 2)
\]

### Step 3: Simplify the fraction
Now, we substitute the factorized form of \(k(x)\) into the original expression:
\[
\frac{x + 2}{(x - 8)(x + 2)}
\]

We can cancel out the common factor of \(x + 2\) from the numerator and denominator, leaving us with:
\[
\frac{1}{x - 8}, \quad \text{for} \, x \neq -2
\]

The solution is:
\[
\frac{f(x)}{k(x)} = \frac{1}{x - 8}, \, \text{as long as} \, x \neq -2
\]

Would you like a further breakdown or have any questions?

### Related questions:
1. How would you perform \( \frac{g(x)}{h(x)} \) using similar methods?
2. What happens if \( x = -2 \) in this problem?
3. How do you factor other quadratic expressions like \(x^2 - 6x - 16\)?
4. Could you use synthetic division instead of factoring here?
5. How would you graph both \(f(x)\) and \(k(x)\) to understand the solution better?

### Tip:
When simplifying fractions, always check for values of \(x\) that may cause the denominator to be zero to avoid undefined solutions.

Find \( \frac{f(x)}{k(x)} \) where \( f(x) = x + 2 \) and \( k(x) = x^2 - 6x - 16 \).

Learn how to divide the functions \(f(x) = x + 2\) and \(k(x) = x^2 - 6x - 16\) by simplifying rational expressions through factoring. This step-by-step solution covers core algebraic concepts like factoring quadratics and simplifying fractions, suitable for students in Grades 8-10.

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