`(3a-2b+5c)^(2)`

Answer

To find the square of the expression \((3a - 2b + 5c)^2\), we will use the algebraic identity for the square of a trinomial, which states:

\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac \]

In our case, we will substitute \(a\) with \(3a\), \(b\) with \(-2b\), and \(c\) with \(5c\).

### Step-by-Step Solution:

1. **Identify the terms**:
- Let \(x = 3a\)
- Let \(y = -2b\)
- Let \(z = 5c\)

2. **Apply the identity**:
\[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx \]

3. **Calculate \(x^2\)**:
\[ x^2 = (3a)^2 = 9a^2 \]

4. **Calculate \(y^2\)**:
\[ y^2 = (-2b)^2 = 4b^2 \]

5. **Calculate \(z^2\)**:
\[ z^2 = (5c)^2 = 25c^2 \]

6. **Calculate \(2xy\)**:
\[ 2xy = 2 \cdot (3a) \cdot (-2b) = -12ab \]

7. **Calculate \(2yz\)**:
\[ 2yz = 2 \cdot (-2b) \cdot (5c) = -20bc \]

8. **Calculate \(2zx\)**:
\[ 2zx = 2 \cdot (5c) \cdot (3a) = 30ac \]

9. **Combine all the results**:
\[ (3a - 2b + 5c)^2 = 9a^2 + 4b^2 + 25c^2 - 12ab - 20bc + 30ac \]

### Final Answer:
\[ (3a - 2b + 5c)^2 = 9a^2 + 4b^2 + 25c^2 - 12ab - 20bc + 30ac \]