Consider all possible integers $n \ge 0$ such that $(5 \cdot 3^m) + 4 = n^2$ holds for some corresponding integer $m \ge 0$. Find the sum of all such $n$.
1. Start with the given equation:
   \[
   5 \cdot 3^m + 4 = n^2
   \]
   Rearrange it to:
   \[
   5 \cdot 3^m = n^2 - 4
   \]
   Notice that \( n^2 - 4 \) can be factored as:
   \[
   n^2 - 4 = (n-2)(n+2)
   \]
   Therefore, we have:
   \[
   5 \cdot 3^m = (n-2)(n+2)
   \]

2. Since \( n-2 \) and \( n+2 \) are two factors of \( 5 \cdot 3^m \), and they differ by 4, we need to consider the possible factorizations of \( 5 \cdot 3^m \) into two factors that differ by 4.

3. We analyze the gcd condition:
   \[
   \gcd(n-2, n+2) = \gcd(n-2, 4) = 1 \text{ or } 2 \text{ or } 4
   \]
   Since \( n-2 \) and \( n+2 \) are both odd, their gcd must be 1. This means we need to consider the following cases:

4. Case 1:
   \[
   n-2 = 5 \quad \text{and} \quad n+2 = 3^m
   \]
   Solving for \( n \):
   \[
   n - 2 = 5 \implies n = 7
   \]
   \[
   n + 2 = 3^m \implies 7 + 2 = 3^m \implies 3^m = 9 \implies m = 2
   \]
   Thus, one solution is \( (n, m) = (7, 2) \).

5. Case 2:
   \[
   n-2 = 3^m \quad \text{and} \quad n+2 = 5
   \]
   Solving for \( n \):
   \[
   n + 2 = 5 \implies n = 3
   \]
   \[
   n - 2 = 3^m \implies 3 - 2 = 3^m \implies 3^m = 1 \implies m = 0
   \]
   Thus, another solution is \( (n, m) = (3, 0) \).

6. Case 3:
   \[
   n-2 = 1 \quad \text{and} \quad n+2 = 5 \cdot 3^m
   \]
   Solving for \( n \):
   \[
   n - 2 = 1 \implies n = 3
   \]
   \[
   n + 2 = 5 \cdot 3^m \implies 3 + 2 = 5 \cdot 3^m \implies 5 = 5 \cdot 3^m \implies 3^m = 1 \implies m = 0
   \]
   This case does not provide a new solution since \( (n, m) = (3, 0) \) is already found.

7. Case 4:
   \[
   n-2 = 5 \cdot 3^m \quad \text{and} \quad n+2 = 1
   \]
   Solving for \( n \):
   \[
   n + 2 = 1 \implies n = -1
   \]
   \[
   n - 2 = 5 \cdot 3^m \implies -1 - 2 = 5 \cdot 3^m \implies -3 = 5 \cdot 3^m
   \]
   This case does not provide integer solutions since \( 5 \cdot 3^m \) is always positive.

8. Summarize the solutions:
   The valid solutions are \( (n, m) = (7, 2) \) and \( (3, 0) \).

9. Calculate the sum of all such \( n \):
   \[
   7 + 3 = 10
   \]

The final answer is \(\boxed{10}\).