Certainly! Let's work through the problem step-by-step to factor the expression [tex]\( 24a^3 + 32a^2 + 40a \)[/tex] using the greatest common factor (GCF).
### Step 1: Identify the Greatest Common Factor (GCF)
First, we need to determine the GCF of the coefficients (24, 32, and 40) and the variable terms [tex]\(a^3\)[/tex], [tex]\(a^2\)[/tex], and [tex]\(a\)[/tex].
- The GCF of the coefficients 24, 32, and 40 is:
- Prime factorization of 24: [tex]\(24 = 2^3 \cdot 3\)[/tex]
- Prime factorization of 32: [tex]\(32 = 2^5\)[/tex]
- Prime factorization of 40: [tex]\(40 = 2^3 \cdot 5\)[/tex]
- The common factor is [tex]\(2^3 = 8\)[/tex].
- The variable terms are [tex]\(a^3\)[/tex], [tex]\(a^2\)[/tex], and [tex]\(a\)[/tex].
The GCF for these terms is [tex]\(a\)[/tex] (since [tex]\(a\)[/tex] is the lowest power common to all terms).
So, the overall GCF is [tex]\(8a\)[/tex].
### Step 2: Factor out the GCF
Next, we factor out [tex]\(8a\)[/tex] from each term in the expression:
[tex]\[ 24a^3 + 32a^2 + 40a \][/tex]
We divide each term by [tex]\(8a\)[/tex]:
[tex]\[
\begin{align*}
24a^3 & = 8a \cdot (3a^2) \\
32a^2 & = 8a \cdot (4a) \\
40a & = 8a \cdot 5
\end{align*}
\][/tex]
Now, we can write the factored expression as:
[tex]\[ 24a^3 + 32a^2 + 40a = 8a(3a^2 + 4a + 5) \][/tex]
### Step 3: Verify the Options
Now, we compare our factored expression with the given multiple-choice options:
A. [tex]\(8(3a^3 + 4a^2 + 5a)\)[/tex]
B. [tex]\(8a(3a^2 + 5a + 4)\)[/tex]
C. [tex]\(8(3a^3 + 5a^2 + 4)\)[/tex]
D. [tex]\(8a(3a^2 + 4a + 5)\)[/tex]
The correct factorization is:
[tex]\[ 8a(3a^2 + 4a + 5) \][/tex]
### Conclusion
Therefore, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
