To determine which of the given expressions is equivalent to the given area of the hexagon [tex]\(24a^2 - 18\)[/tex] square units, let's compare each option step-by-step by simplifying them and see which simplifies to [tex]\(24a^2 - 18\)[/tex].
### Option 1: [tex]\( 6(4a^2 - 3) \)[/tex]
Let's distribute the 6:
[tex]\[ 6 \times (4a^2 - 3) = 6 \times 4a^2 - 6 \times 3 \][/tex]
[tex]\[ = 24a^2 - 18 \][/tex]
### Option 2: [tex]\( 6(8a^2 - 9) \)[/tex]
Let's distribute the 6:
[tex]\[ 6 \times (8a^2 - 9) = 6 \times 8a^2 - 6 \times 9 \][/tex]
[tex]\[ = 48a^2 - 54 \][/tex]
### Option 3: [tex]\( 6a(12a - 9) \)[/tex]
Let's distribute the [tex]\(6a\)[/tex]:
[tex]\[ 6a \times (12a - 9) = 6a \times 12a - 6a \times 9 \][/tex]
[tex]\[ = 72a^2 - 54a \][/tex]
### Option 4: [tex]\( 6a(18a - 12) \)[/tex]
Let's distribute the [tex]\(6a\)[/tex]:
[tex]\[ 6a \times (18a - 12) = 6a \times 18a - 6a \times 12 \][/tex]
[tex]\[ = 108a^2 - 72a \][/tex]
Now, compare the simplified forms of each option with the given expression [tex]\(24a^2 - 18\)[/tex]:
- Option 1: [tex]\( 24a^2 - 18 \)[/tex] matches exactly.
- Option 2: [tex]\( 48a^2 - 54 \)[/tex] does not match.
- Option 3: [tex]\( 72a^2 - 54a \)[/tex] does not match.
- Option 4: [tex]\( 108a^2 - 72a \)[/tex] does not match.
Thus, the equivalent expression for the area of the hexagon based on the area of a triangle is:
[tex]\[ \boxed{6(4a^2 - 3)} \][/tex]
