To determine which of the given choices is equivalent to the expression [tex]\(\frac{1}{4}(8 - 6x + 12)\)[/tex], we need to simplify the expression step by step.
1. Combine like terms inside the parentheses if possible:
[tex]\[
8 - 6x + 12
\][/tex]
Combine [tex]\(8\)[/tex] and [tex]\(12\)[/tex]:
[tex]\[
8 + 12 = 20
\][/tex]
Thus, the expression inside the parentheses simplifies to:
[tex]\[
20 - 6x
\][/tex]
2. Distribute [tex]\(\frac{1}{4}\)[/tex] to each term inside the parentheses:
[tex]\[
\frac{1}{4}(20 - 6x)
\][/tex]
Distribute [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\frac{1}{4} \cdot 20 - \frac{1}{4} \cdot 6x
\][/tex]
3. Perform the multiplications:
[tex]\[
\frac{1}{4} \cdot 20 = \frac{20}{4} = 5
\][/tex]
and
[tex]\[
\frac{1}{4} \cdot 6x = \frac{6x}{4} = \frac{3x}{2}
\][/tex]
4. Combine the simplified terms:
[tex]\[
5 - \frac{3x}{2}
\][/tex]
5. Match the simplified expression with the choices given:
[tex]\[
5 - \frac{3x}{2}
\][/tex]
This matches with choice D, which is:
[tex]\[
D. -\frac{3}{2} x + 5
\][/tex]
Thus, the expression equivalent to [tex]\(\frac{1}{4}(8 - 6x + 12)\)[/tex] is [tex]\(\boxed{-\frac{3}{2} x + 5}\)[/tex].
