To determine which expressions are equivalent to [tex]\(4d + 6 + 2d\)[/tex], we should first simplify the given expression and then compare it to the provided options. Let's go through the steps one by one:
1. Simplify the original expression [tex]\(4d + 6 + 2d\)[/tex]:
[tex]\[
4d + 6 + 2d = (4d + 2d) + 6 = 6d + 6
\][/tex]
2. Check each option for equivalence:
Option A: [tex]\(2(3d + 3)\)[/tex]
Simplify the expression:
[tex]\[
2(3d + 3) = 2 \cdot 3d + 2 \cdot 3 = 6d + 6
\][/tex]
Since [tex]\(6d + 6\)[/tex] is equivalent to the simplified original expression [tex]\(6d + 6\)[/tex], this option is equivalent.
Option B: [tex]\(6(d + 6)\)[/tex]
Simplify the expression:
[tex]\[
6(d + 6) = 6 \cdot d + 6 \cdot 6 = 6d + 36
\][/tex]
Since [tex]\(6d + 36\)[/tex] is not equivalent to the simplified original expression [tex]\(6d + 6\)[/tex], this option is not equivalent.
Option C: [tex]\((3d + 3) + (3d + 3)\)[/tex]
Simplify the expression:
[tex]\[
(3d + 3) + (3d + 3) = 3d + 3 + 3d + 3 = (3d + 3d) + (3 + 3) = 6d + 6
\][/tex]
Since [tex]\(6d + 6\)[/tex] is equivalent to the simplified original expression [tex]\(6d + 6\)[/tex], this option is equivalent.
Therefore, the expressions that are equivalent to [tex]\(4d + 6 + 2d\)[/tex] are:
- A [tex]\(2(3d + 3)\)[/tex]
- C [tex]\((3d + 3) + (3d + 3)\)[/tex]
So, the correct answers are A and C.
