The given expression is:

\[
2a^3 + 36a^2 + 162a
\]

This expression can be simplified by factoring out the greatest common factor (GCF). Let's factor it:

1. Identify the GCF of the terms: \(2a\).
2. Factor \(2a\) out of the expression:

\[
2a(a^2 + 18a + 81)
\]

Now, let's check if the quadratic expression inside the parentheses can be factored further. The quadratic expression is \(a^2 + 18a + 81\).

3. Factor the quadratic expression:

\[
a^2 + 18a + 81 = (a + 9)(a + 9) = (a + 9)^2
\]

4. Substitute back into the original expression:

\[
2a(a + 9)^2
\]

So, the fully factored form of the expression is:

\[
2a(a + 9)^2
\]

Would you like more details or have any questions about this factorization?

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Here are 5 related questions to expand your understanding:

1. How do you factor a quadratic trinomial?
2. What are the conditions for a quadratic trinomial to be a perfect square?
3. How can you factor polynomials with higher degrees?
4. What is the role of the greatest common factor in simplifying expressions?
5. Can the expression \(2a(a + 9)^2\) be expanded back into its original form?

**Tip:** Always look for the greatest common factor first when simplifying algebraic expressions; it can make the rest of the factorization process easier.

Learn how to factorize the algebraic expression 2a^3 + 36a^2 + 162a step-by-step. This problem covers key algebraic concepts, including factoring polynomials and the factor theorem. Suitable for high school students.

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