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We are given the quadratic expression:

\[
(x - 11)^2 + 5(x - 11) - 24
\]

Let's factor it completely. To make this process easier, we can first use substitution. Let:

\[
u = x - 11
\]

This turns the original equation into:

\[
u^2 + 5u - 24
\]

Now we can factor this quadratic expression. We need two numbers that multiply to \(-24\) and add up to \(5\). These numbers are \(8\) and \(-3\), so we can rewrite the expression as:

\[
u^2 + 8u - 3u - 24
\]

Group the terms:

\[
(u^2 + 8u) - (3u + 24)
\]

Factor each group:

\[
u(u + 8) - 3(u + 8)
\]

Now, factor out the common binomial \((u + 8)\):

\[
(u - 3)(u + 8)
\]

Finally, substitute back \(u = x - 11\) to get the fully factored form:

\[
(x - 11 - 3)(x - 11 + 8) = (x - 14)(x - 3)
\]

So, the factored form of the original expression is:

\[
(x - 14)(x - 3)
\]

Would you like a detailed explanation of any part of this? 

Here are five follow-up questions that build on this:

1. What would the expanded form of \((x - 14)(x - 3)\) look like?
2. Can you explain how the method of substitution simplifies factoring?
3. How do you identify which numbers to choose when factoring quadratic expressions?
4. What is the significance of factoring completely in algebraic expressions?
5. What other methods can be used to factor more complex quadratic expressions?

**Tip:** When factoring, always look for a common substitution or a method to simplify the expression before proceeding with factoring steps.

Factor completely: (x - 11)^2 + 5(x - 11) - 24

This solution shows how to factor the quadratic expression (x - 11)^2 + 5(x - 11) - 24 using substitution and common factorization techniques. Step-by-step explanation suitable for Grades 9-11.

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