To find the value of the expression \(\frac{1}{4}\left(c^3 + d^2\right)\) when \(c = -4\) and \(d = 10\), follow these steps:
1. Substitute the values of \(c\) and \(d\) into the expression:
[tex]\[
\frac{1}{4}\left((-4)^3 + 10^2\right)
\][/tex]
2. Calculate \((-4)^3\):
[tex]\[
(-4)^3 = (-4) \times (-4) \times (-4) = -64
\][/tex]
3. Calculate \(10^2\):
[tex]\[
10^2 = 10 \times 10 = 100
\][/tex]
4. Add these results together:
[tex]\[
(-64) + 100 = 36
\][/tex]
5. Multiply the sum by \(\frac{1}{4}\):
[tex]\[
\frac{1}{4} \times 36 = 9
\][/tex]
Therefore, the value of the expression \(\frac{1}{4}\left(c^3 + d^2\right)\) when \(c = -4\) and \(d = 10\) is \(9\).
Hence, the correct answer is [tex]\( \boxed{9} \)[/tex].
