To determine $f(k+3)$ for the function $f(x) = x^2 - 8x - 12$, we need to substitute $k+3$ in place of $x$ in the given function:

$f(k+3) = (k+3)^2 - 8(k+3) - 12$

Now, let's simplify this expression step by step:

Step 1: Expand $(k+3)^2$

$(k+3)^2 = k^2 + 6k + 9$

Step 2: Expand $-8(k+3)$

$-8(k+3) = -8k - 24$

Step 3: Substitute and combine like terms

Now substitute these expansions into the original expression:

$f(k+3) = (k^2 + 6k + 9) - 8k - 24 - 12$

Simplify:

$f(k+3) = k^2 + 6k + 9 - 8k - 24 - 12$ $f(k+3) = k^2 - 2k - 27$

Thus, the simplified expression for $f(k+3)$ is:

$f(k+3) = k^2 - 2k - 27$

Would you like to go over any step in more detail or have other questions? Here's a list of related questions to help you explore the topic further:

1. How do we factor a quadratic function like $k^2 - 2k - 27$?
2. What are the solutions to the equation $k^2 - 2k - 27 = 0$?
3. How can we complete the square for the function $f(x) = x^2 - 8x - 12$?
4. What does the vertex of the quadratic function $f(x) = x^2 - 8x - 12$ represent?
5. How does the graph of $f(x) = x^2 - 8x - 12$ change when we replace $x$ with $x + 3$?

Tip: When working with quadratic functions, always remember that factoring and completing the square are powerful techniques for solving and simplifying expressions!