To solve this problem, we need to find an equivalent expression for the area of the hexagon given by [tex]\(24a^2 - 18\)[/tex] square units. Let's start by factoring out the greatest common factor (GCF) from the given expression.
1. Identify the terms in the expression:
The given expression is:
[tex]\[
24a^2 - 18
\][/tex]
2. Look for the GCF among the coefficients (24 and 18) and the variable parts:
The coefficient of [tex]\(a^2\)[/tex] is 24, and the constant term is 18. Both coefficients (24 and 18) share a common factor.
3. Determine the greatest common factor (GCF):
- The factors of 24 are: [tex]\(1, 2, 3, 4, 6, 8, 12, 24\)[/tex]
- The factors of 18 are: [tex]\(1, 2, 3, 6, 9, 18\)[/tex]
- The greatest common factor (GCF) of 24 and 18 is 6.
4. Factor out the GCF from each term:
[tex]\[
24a^2 - 18 = 6(4a^2 - 3)
\][/tex]
Here, we factored out [tex]\(6\)[/tex], and the remaining terms inside the parentheses form the polynomial [tex]\(4a^2 - 3\)[/tex].
5. Verify the factorization:
We can confirm the factorization by expanding [tex]\(6(4a^2 - 3)\)[/tex]:
[tex]\[
6 \cdot 4a^2 = 24a^2
\][/tex]
[tex]\[
6 \cdot (-3) = -18
\][/tex]
Adding these terms together gives us back the original expression:
[tex]\[
24a^2 - 18
\][/tex]
Therefore, the equivalent expression for the area of the hexagon is [tex]\(6(4a^2 - 3)\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{6(4a^2 - 3)}
\][/tex]
