To determine which expressions are equivalent to [tex]\(3(-2a - 4) + 3a\)[/tex], let's go through a detailed, step-by-step process of simplifying the given expression and comparing it to each of the options.
### Step 1: Simplify the given expression
Let's start by simplifying the expression [tex]\(3(-2a - 4) + 3a\)[/tex]:
1. Distribute the 3:
[tex]\[
3(-2a - 4) + 3a = 3 \cdot (-2a) + 3 \cdot (-4) + 3a
\][/tex]
2. Simplify each term:
[tex]\[
3 \cdot (-2a) = -6a
\][/tex]
[tex]\[
3 \cdot (-4) = -12
\][/tex]
Now, substitute these back into the expression:
[tex]\[
-6a - 12 + 3a
\][/tex]
3. Combine like terms:
[tex]\[
-6a + 3a - 12
\][/tex]
[tex]\[
(-6a + 3a) - 12 = -3a - 12
\][/tex]
Thus, the simplified form of the expression [tex]\(3(-2a - 4) + 3a\)[/tex] is:
[tex]\[
-3a - 12
\][/tex]
### Step 2: Compare with each option
Now we will compare the simplified expression [tex]\(-3a - 12\)[/tex] to each of the given options:
- Option A: [tex]\(-6a - 12 + 3a\)[/tex]
Simplify Option A:
[tex]\[
-6a - 12 + 3a = (-6a + 3a) - 12 = -3a - 12
\][/tex]
This is exactly [tex]\(-3a - 12\)[/tex], which matches our simplified expression.
- Option B: [tex]\(3a + 12\)[/tex]
[tex]\(3a + 12\)[/tex] is not equivalent to [tex]\(-3a - 12\)[/tex]. The signs and terms do not match, so this expression is incorrect.
### Conclusion
Based on the above steps:
- Option A [tex]\(-6a - 12 + 3a\)[/tex] simplifies to [tex]\(-3a - 12\)[/tex] and is equivalent to the given expression.
- Option B [tex]\(3a + 12\)[/tex] does not match [tex]\(-3a - 12\)[/tex].
So, the correct answer is:
A. [tex]\(-6a - 12 + 3a\)[/tex]
