To expand the expression \((x + 9)(x + 3)(x + 10)\), we'll proceed step by step.

### Step 1: Expand the first two factors
First, expand \((x + 9)(x + 3)\):

\[
(x + 9)(x + 3) = x(x + 3) + 9(x + 3) = x^2 + 3x + 9x + 27 = x^2 + 12x + 27
\]

### Step 2: Multiply the result by the third factor
Now, multiply \((x^2 + 12x + 27)\) by \((x + 10)\):

\[
(x^2 + 12x + 27)(x + 10) = x^2(x + 10) + 12x(x + 10) + 27(x + 10)
\]

### Step 3: Expand each term
Expand each term individually:

\[
x^2(x + 10) = x^3 + 10x^2
\]
\[
12x(x + 10) = 12x^2 + 120x
\]
\[
27(x + 10) = 27x + 270
\]

### Step 4: Combine like terms
Now, combine all the expanded terms:

\[
x^3 + 10x^2 + 12x^2 + 120x + 27x + 270 = x^3 + (10x^2 + 12x^2) + (120x + 27x) + 270
\]
\[
= x^3 + 22x^2 + 147x + 270
\]

So, the expanded form of \((x + 9)(x + 3)(x + 10)\) is:

\[
\boxed{x^3 + 22x^2 + 147x + 270}
\]

Would you like more details or have any questions?  

Here are some related questions to consider:

1. How would the expansion change if one of the terms was negative?
2. What is the significance of the coefficients in the expanded polynomial?
3. How do the roots of the original expression relate to the expanded form?
4. Can this expanded polynomial be factored back into its original binomials?
5. What are the applications of polynomial expansion in real-world problems?
6. How does the degree of the polynomial affect its graph?
7. How would you handle the expansion if one of the binomials was a square term?
8. How can synthetic division be used to verify the factors of the expanded polynomial?

**Tip:** Always double-check your work by multiplying the expanded terms to ensure the correctness of your expansion.

Learn how to expand the polynomial (x + 9)(x + 3)(x + 10) with a detailed step-by-step solution. This problem covers polynomial expansion using the binomial theorem and is suitable for advanced high school students. Understand the significance of coefficients and related applications in real-world problems.

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