To determine which expressions are equivalent to [tex]\(6g - 18h\)[/tex], we need to expand and simplify each option and compare it to [tex]\(6g - 18h\)[/tex]:
Option A: [tex]\((g - 3) \cdot 6\)[/tex]
Expanding this, we get:
[tex]\[
(g - 3) \cdot 6 = 6 \cdot g - 6 \cdot 3 = 6g - 18
\][/tex]
This is not equivalent to [tex]\(6g - 18h\)[/tex] as the terms involving [tex]\(h\)[/tex] are different.
Option B: [tex]\(2 \cdot (3g - 18h)\)[/tex]
Expanding this, we get:
[tex]\[
2 \cdot (3g - 18h) = 2 \cdot 3g - 2 \cdot 18h = 6g - 36h
\][/tex]
This is not equivalent to [tex]\(6g - 18h\)[/tex] as the coefficient of [tex]\(h\)[/tex] is different.
Option C: [tex]\(3(2g - 6h)\)[/tex]
Expanding this, we get:
[tex]\[
3(2g - 6h) = 3 \cdot 2g - 3 \cdot 6h = 6g - 18h
\][/tex]
This is equivalent to [tex]\(6g - 18h\)[/tex].
Option D: [tex]\((-g - 3h)(-6)\)[/tex]
Expanding this, we get:
[tex]\[
(-g - 3h)(-6) = (-g) \cdot (-6) + (-3h) \cdot (-6) = 6g + 18h
\][/tex]
This is not equivalent to [tex]\(6g - 18h\)[/tex] as the sign in front of the [tex]\(h\)[/tex]-term is different.
Option E: [tex]\(-2 \times (-3g + 9h)\)[/tex]
Expanding this, we get:
[tex]\[
-2 \times (-3g + 9h) = -2 \cdot (-3g) + (-2) \cdot 9h = 6g - 18h
\][/tex]
This is equivalent to [tex]\(6g - 18h\)[/tex].
Thus, the expressions that are equivalent to [tex]\(6g - 18h\)[/tex] are:
C. [tex]\(3(2g - 6h)\)[/tex] and E. [tex]\(-2 \times (-3g + 9h)\)[/tex]
