The expression is:

$$
\(3 c+d^2\)
$$


To find the greatest possible value, we need to independently maximize each term in the expression.
1. Maximize the $3 c$ Term

Since $\(3 c\)$ is a positive multiple of $c$, we must choose the largest possible value for $c$.

Possible values for $\(c:\{3,-4,-12\}\)$
The largest value for $c$ is 3 .

$$
\(\text { Maximum } 3 c=3 \times 3=9\)
$$

2. Maximize the $\(d^2\)$ Term

Since $\(d^2\)$ is a squared term, it will always be positive (or zero). To maximize $\(d^2\)$, we must choose the value for $d$ that has the largest absolute value.

Possible values for $\(d:\{-5,4,2\}\)$

We calculate $d^2$ for each possibility:
- $\((-5)^2=25\)$
- $\((4)^2=16\)$
- $\((2)^2=4\)$

The largest value for $\(d^2\)$ is $\(\mathbf{2 5}\)$.
3. Calculate the Greatest Possible Value

Now we combine the maximum values for both terms to find the greatest possible value of the entire expression:

$$
\(\begin{gathered}
\text { Greatest Value }=(\text { Maximum } 3 c)+\left(\text { Maximum } d^2\right) \\
\text { Greatest Value }=9+25=34
\end{gathered}\)
$$


The greatest possible value for $\(3 c+d^2\)$ is $\(\mathbf{3 4}\)$.