We can factor the expression \( 4a^8 - 12a^7 - 216a^6 \). 

### Step 1: Factor out the greatest common factor (GCF)
The GCF of all the terms is \( 4a^6 \), as 4 is the largest number that divides all the coefficients, and \( a^6 \) is the highest power of \( a \) common to all the terms. Let's factor that out:

\[
4a^6(a^2 - 3a - 54)
\]

### Step 2: Factor the quadratic expression
Now, we need to check if the quadratic expression \( a^2 - 3a - 54 \) can be factored. We are looking for two numbers that multiply to \( -54 \) and add up to \( -3 \).

After checking, we find that it cannot be factored further using integers. Therefore, the fully factored form is:

\[
4a^6(a^2 - 3a - 54)
\]

This is the final factorization.

Would you like more details or have any questions?

### Related Questions:
1. How do you determine the greatest common factor of polynomials?
2. Can the quadratic formula solve \( a^2 - 3a - 54 \)?
3. How do you check if a quadratic expression can be factored?
4. What happens when a quadratic cannot be factored using integers?
5. How do you solve quadratic equations using the quadratic formula?

### Tip:
Always start by factoring out the greatest common factor (GCF) to simplify the polynomial before looking for further factorization.

Learn how to factor the polynomial 4a^8 - 12a^7 - 216a^6 by factoring out the greatest common factor (GCF). This step-by-step solution simplifies the expression to 4a^6(a^2 - 3a - 54). Perfect for students in Grades 9-12 studying polynomial factorization and algebra.

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