https://file.aiquickdraw.com/mathbot/file/1726847162847nujnqv3m.jpg

The expression in the image is: 

\[
54a^4 + 432ab^3
\]

Let's factor this expression fully.

### Step 1: Identify the greatest common factor (GCF)
The common factor between the two terms is \(54a\):
- \(54a^4 = 54a \cdot a^3\)
- \(432ab^3 = 54a \cdot 8b^3\)

### Step 2: Factor out the GCF
\[
54a(a^3 + 8b^3)
\]

### Step 3: Factor the remaining expression
The expression inside the parentheses, \(a^3 + 8b^3\), is a sum of cubes:
\[
a^3 + 8b^3 = a^3 + (2b)^3
\]
This can be factored using the sum of cubes formula:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
Applying this formula:
\[
a^3 + (2b)^3 = (a + 2b)(a^2 - 2ab + 4b^2)
\]

### Final Factored Form
Thus, the fully factored form of the expression is:
\[
54a(a + 2b)(a^2 - 2ab + 4b^2)
\]

Would you like further details on any step? Here are some related questions for practice:

1. How do you find the greatest common factor (GCF)?
2. What is the formula for factoring the sum of cubes?
3. How do you factor the difference of cubes?
4. How do you check if an expression is completely factored?
5. How can you apply factoring techniques to solve equations?

**Tip:** When factoring, always start by looking for a common factor across all terms!

Learn how to fully factor the algebraic expression 54a^4 + 432ab^3 by identifying the greatest common factor (GCF) and applying the sum of cubes formula. This step-by-step solution is suitable for students in Grades 9-12 and covers important algebraic techniques.

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