Three positive reals $x , y , z $ satisfy \\
$x^2 + y^2 = 3^2 \\
y^2 + yz + z^2 = 4^2 \\
x^2 + \sqrt{3}xz + z^2 = 5^2 .$ \\
Find the value of $2xy + xz + \sqrt{3}yz$
Given the equations:
\[ x^2 + y^2 = 3^2 \]
\[ y^2 + yz + z^2 = 4^2 \]
\[ x^2 + \sqrt{3}xz + z^2 = 5^2 \]

We need to find the value of \( 2xy + xz + \sqrt{3}yz \).

1. **Rewrite the equations using the given values:**
   \[ x^2 + y^2 = 9 \]
   \[ y^2 + yz + z^2 = 16 \]
   \[ x^2 + \sqrt{3}xz + z^2 = 25 \]

2. **Use the Cosine Rule to interpret the equations geometrically:**
   The Cosine Rule states that for any triangle with sides \(a\), \(b\), and \(c\), and the angle \(\theta\) opposite side \(c\):
   \[ c^2 = a^2 + b^2 - 2ab\cos(\theta) \]

3. **Identify the angles corresponding to the given equations:**
   - For \( x^2 + y^2 = 9 \), we can interpret this as:
     \[ x^2 + y^2 - 2xy\cos(90^\circ) = 9 \]
     Since \(\cos(90^\circ) = 0\), this simplifies to:
     \[ x^2 + y^2 = 9 \]

   - For \( y^2 + yz + z^2 = 16 \), we can interpret this as:
     \[ y^2 + z^2 - 2yz\cos(120^\circ) = 16 \]
     Since \(\cos(120^\circ) = -\frac{1}{2}\), this simplifies to:
     \[ y^2 + z^2 + yz = 16 \]

   - For \( x^2 + \sqrt{3}xz + z^2 = 25 \), we can interpret this as:
     \[ x^2 + z^2 - 2xz\cos(150^\circ) = 25 \]
     Since \(\cos(150^\circ) = -\frac{\sqrt{3}}{2}\), this simplifies to:
     \[ x^2 + z^2 + \sqrt{3}xz = 25 \]

4. **Recognize the geometric interpretation:**
   The equations suggest that \(x\), \(y\), and \(z\) are the sides of triangles formed by an interior point of a \(3-4-5\) right triangle.

5. **Calculate the required expression:**
   The expression \(2xy + xz + \sqrt{3}yz\) can be interpreted as the sum of the areas of the triangles formed by the sides \(3, x, y\); \(4, y, z\); and \(5, x, z\).

6. **Equate the area of the \(3-4-5\) triangle:**
   The area of the \(3-4-5\) right triangle is:
   \[ \text{Area} = \frac{1}{2} \times 3 \times 4 = 6 \]

   The sum of the areas of the triangles formed by the sides \(3, x, y\); \(4, y, z\); and \(5, x, z\) is:
   \[ 2xy + xz + \sqrt{3}yz = 24 \]

Thus, the value of \(2xy + xz + \sqrt{3}yz\) is \(\boxed{24}\).