To factor the polynomial \( 81x^4 + 72ax^3 - 12a^2x^2 - 16xa^3 \), we can approach it by grouping the terms.

### Step 1: Group the terms in pairs
Group the polynomial in pairs to factor by grouping:
\[
(81x^4 + 72ax^3) - (12a^2x^2 + 16xa^3)
\]

### Step 2: Factor out the common factor from each group
Factor out the greatest common factor (GCF) from each pair:
\[
9x^3(9x + 8a) - 4a^2x(3x + 4a)
\]

### Step 3: Factor by grouping
Now, observe that the expressions inside the parentheses are not the same. However, let's check if a simple factorization is possible:

\[
9x^3(9x + 8a) - 4a^2x(3x + 4a)
\]

The factors aren't directly the same, but let's check if we can factor further or use a common factor:

\[
(9x)(9x^3 + 8ax^2) - 4a(3x + 4a)x
\]

Unfortunately, it seems there's no simpler factorization. Therefore, the most factored form based on the original expression is:

\[
9x^3(9x + 8a) - 4a^2x(3x + 4a)
\]

This is the factorization of the given polynomial.

Would you like more details or have any questions?

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Here are 8 related questions to expand your understanding:

1. How do you factor a polynomial by grouping?
2. What is the greatest common factor (GCF) and how do you find it?
3. Can you explain why certain polynomials are not factorable over the integers?
4. How can you check if a polynomial is factorable?
5. What are other methods to factor complex polynomials?
6. How do you simplify expressions with multiple variables?
7. What are the steps to solve polynomial equations after factoring?
8. How do you know when a polynomial is completely factored?

**Tip:** Always double-check your factorizations by expanding the factors to ensure they multiply back to the original expression.

Learn how to factor the polynomial 81x^4 + 72ax^3 - 12a^2x^2 - 16xa^3 step-by-step with detailed explanations. This problem covers polynomial factorization techniques, including grouping terms and finding the greatest common factor (GCF). Suitable for students in Grades 10-12.

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