Let's simplify the expression:

\[
(x^2 + 7x + 18 + \frac{42}{x-3})(x-3)
\]

Step 1: Distribute \((x - 3)\) across all the terms inside the parentheses.

\[
(x^2 + 7x + 18)(x-3) + \frac{42}{x-3}(x-3)
\]

Step 2: Simplify the second part. Since \(\frac{42}{x-3}(x-3)\) equals 42 (because \(x-3\) cancels out), we are left with:

\[
(x^2 + 7x + 18)(x-3) + 42
\]

Step 3: Now, expand \((x^2 + 7x + 18)(x-3)\):

\[
x^2(x-3) + 7x(x-3) + 18(x-3)
\]

Expanding each term:

\[
x^2(x-3) = x^3 - 3x^2
\]
\[
7x(x-3) = 7x^2 - 21x
\]
\[
18(x-3) = 18x - 54
\]

Step 4: Now, combine all these terms together:

\[
x^3 - 3x^2 + 7x^2 - 21x + 18x - 54 + 42
\]

Step 5: Simplify by combining like terms:

\[
x^3 + 4x^2 - 3x - 12
\]

So, the simplified expression is:

\[
x^3 + 4x^2 - 3x - 12
\]

Would you like further clarification on any of the steps?

Here are 5 related questions to explore further:

1. How do you expand polynomials in general?
2. What is the process for factoring a cubic expression?
3. How do you handle rational expressions during distribution?
4. What are common mistakes when distributing and expanding expressions?
5. How can you verify the correctness of a polynomial simplification?

**Tip:** When distributing terms, carefully expand each product and watch for sign errors to avoid mistakes!

This detailed step-by-step guide shows how to simplify the expression (x^2 + 7x + 18 + 42/x-3)(x-3) using polynomial distribution and rational expression simplification. Ideal for high school algebra students, this problem introduces key concepts like expanding polynomials and simplifying rational terms.

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