Let's break down the problem step by step to find the value of the expression [tex]\(\frac{1}{4} \left( c^3 + d^2 \right)\)[/tex] when [tex]\(c = -4\)[/tex] and [tex]\(d = 10\)[/tex].
1. First, substitute the values of [tex]\(c\)[/tex] and [tex]\(d\)[/tex] into the expression [tex]\(c^3 + d^2\)[/tex]:
[tex]\[
(-4)^3 + 10^2
\][/tex]
2. Calculate [tex]\((-4)^3\)[/tex]:
[tex]\[
(-4) \times (-4) \times (-4) = -64
\][/tex]
3. Calculate [tex]\(10^2\)[/tex]:
[tex]\[
10 \times 10 = 100
\][/tex]
4. Add the two results together:
[tex]\[
-64 + 100 = 36
\][/tex]
5. Finally, multiply the result by [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\frac{1}{4} \times 36 = 9
\][/tex]
Therefore, the value of the expression is [tex]\(\boxed{9}\)[/tex]. So, the correct answer is:
[tex]\[
\boxed{9}
\][/tex]
