Find all pairs of positive integers $(x,n) $ that are solutions of the equation $3 \cdot 2^x +4 =n^2$.
We are given the equation:
\[ 3 \cdot 2^x + 4 = n^2 \]

First, we rewrite the equation in a different form:
\[ 3 \cdot 2^x = n^2 - 4 \]
\[ 3 \cdot 2^x = (n-2)(n+2) \]

We now consider the factorization of \(3 \cdot 2^x\) as \((n-2)(n+2)\). Since \(3\) is a prime number, one of the factors must be a multiple of \(3\). We will analyze the possible cases for \(n\) modulo \(3\).

### Case 1: \(n \equiv 1 \pmod{3}\)
If \(n = 3k + 1\), then:
\[ n-2 = 3k - 1 \]
\[ n+2 = 3k + 3 \]

Substituting these into the factorization:
\[ 3 \cdot 2^x = (3k - 1)(3k + 3) \]

We simplify:
\[ 3 \cdot 2^x = 3(3k^2 + 2k - 1) \]
\[ 2^x = 3k^2 + 2k - 1 \]

Since \(2^x\) is a power of \(2\), we need \(3k^2 + 2k - 1\) to be a power of \(2\). We will test small values of \(k\):

- For \(k = 1\):
  \[ 3(1)^2 + 2(1) - 1 = 3 + 2 - 1 = 4 = 2^2 \]
  Thus, \(x = 2\) and \(n = 3(1) + 1 = 4\).

- For \(k = 2\):
  \[ 3(2)^2 + 2(2) - 1 = 12 + 4 - 1 = 15 \]
  \(15\) is not a power of \(2\).

- For \(k = 3\):
  \[ 3(3)^2 + 2(3) - 1 = 27 + 6 - 1 = 32 = 2^5 \]
  Thus, \(x = 5\) and \(n = 3(3) + 1 = 10\).

So, we have found solutions \((x, n) = (2, 4)\) and \((x, n) = (5, 10)\).

### Case 2: \(n \equiv 2 \pmod{3}\)
If \(n = 3k + 2\), then:
\[ n-2 = 3k \]
\[ n+2 = 3k + 4 \]

Substituting these into the factorization:
\[ 3 \cdot 2^x = 3k(3k + 4) \]

We simplify:
\[ 3 \cdot 2^x = 3k(3k + 4) \]
\[ 2^x = k(3k + 4) \]

Since \(2^x\) is a power of \(2\), we need \(k(3k + 4)\) to be a power of \(2\). We will test small values of \(k\):

- For \(k = 1\):
  \[ 1(3(1) + 4) = 1 \cdot 7 = 7 \]
  \(7\) is not a power of \(2\).

- For \(k = 2\):
  \[ 2(3(2) + 4) = 2 \cdot 10 = 20 \]
  \(20\) is not a power of \(2\).

- For \(k = 4\):
  \[ 4(3(4) + 4) = 4 \cdot 16 = 64 = 2^6 \]
  Thus, \(x = 6\) and \(n = 3(4) + 2 = 14\).

So, we have found another solution \((x, n) = (6, 14)\).

### Conclusion
The pairs of positive integers \((x, n)\) that are solutions to the equation \(3 \cdot 2^x + 4 = n^2\) are:
\[ \boxed{(2, 4), (5, 10), (6, 14)} \]