First, let's split the compound inequality into two parts:
1. $\(8<-2 n^2+40\)$
2. $\(-2 n^2+40<38\)$

Part 1: Solve $\(8<-2 n^2+40\)$

$$
\(8<-2 n^2+40\)
$$


Subtract 40 from both sides:

$$
\(\begin{gathered}
8-40<-2 n^2 \\
-32<-2 n^2
\end{gathered}\)
$$


Divide both sides by -2 (remember to reverse the inequality sign when dividing by a negative number):

$$
\(\begin{aligned}
& 16>n^2 \\
& n^2<16
\end{aligned}\)
$$


This implies:

$$
\(-4<n<4\)
$$


Part 2: Solve $\(-2 n^2+40<38\)$

$$
\(-2 n^2+40<38\)
$$


Subtract 40 from both sides:

$$
\(-2 n^2<-2\)
$$


Divide both sides by -2 (reverse the inequality sign again):

$$
\(n^2>1\)
$$


This implies:

$$
\(n<-1 \quad \text { or } \quad n>1\)
$$


Step 2: Combine the Results
From Part 1, we have:

$$
\(-4<n<4\)
$$


From Part 2, we have:

$$
\(n<-1 \quad \text { or } \quad n>1\)
$$


Combining these, we get two ranges for $n$ :
1. $\(-4<n<-1\)$
2. $\(1<n<4\)$


Step 3: Find Integer Solutions
Now, let's list the integer values of $n$ that fall within these ranges.
Range 1: $\(-4<n<-1\)$
Possible integer values:

$$
\(n=-3,-2\)
$$


Range 2: $\(1<n<4\)$
Possible integer values:

$$
\(n=2,3\)
$$


Step 4: Verify the Solutions
Let's verify each potential solution in the original inequality to ensure they satisfy both parts.
1. For $\(n=-3\)$ :

$$
\(-2(-3)^2+40=-2(9)+40=-18+40=22
$$\)


Check: $\(8<22<38 \rightarrow\)$ Valid
2. For $\(n=-2\)$ :

$$
\(-2(-2)^2+40=-2(4)+40=-8+40=32\)
$$


Check: $\(8<32<38 \rightarrow\)$ Valid
3. For $\(n=2\)$ :

$$
\(-2(2)^2+40=-2(4)+40=-8+40=32\)
$$


Check: $\(8<32<38 \rightarrow\)$ Valid
4. For $\(n=3\)$ :

$$
\(-2(3)^2+40=-2(9)+40=-18+40=22\)
$$


Check: $\(8<22<38 \rightarrow\)$ Valid
Step 5: Check Boundary Cases
The inequalities are strict (< and >), so we exclude the boundary values:
- $\(n=-4: n^2=16 \rightarrow\)$ Violates $\(n^2<16\)$
- $\(n=-1: n^2=1 \rightarrow\)$ Violates $\(n^2>1\)$
- $\(n=1: n^2=1 \rightarrow\)$ Violates $\(n^2>1\)$
- $\(n=4: n^2=16 \rightarrow\)$ Violates $\(n^2<16\)$

Conclusion
There are four integer values of $n$ that satisfy the inequality: $\(-3,-2,2,3\)$.