Expand: `(3x+ 5y+ 2z) (3x- 5y + 2z)`

Answer

To expand the expression \((3x + 5y + 2z)(3x - 5y + 2z)\), we will use the distributive property (also known as the FOIL method for binomials).

### Step-by-Step Solution:

1. **Write down the expression:**
\[ (3x + 5y + 2z)(3x - 5y + 2z) \]

2. **Distribute each term in the first bracket across the second bracket:**
- Multiply \(3x\) by each term in the second bracket:
\[ 3x \cdot (3x) + 3x \cdot (-5y) + 3x \cdot (2z) = 9x^2 - 15xy + 6xz \]
- Multiply \(5y\) by each term in the second bracket:
\[ 5y \cdot (3x) + 5y \cdot (-5y) + 5y \cdot (2z) = 15xy - 25y^2 + 10yz \]
- Multiply \(2z\) by each term in the second bracket:
\[ 2z \cdot (3x) + 2z \cdot (-5y) + 2z \cdot (2z) = 6xz - 10yz + 4z^2 \]

3. **Combine all the results:**
\[ 9x^2 - 15xy + 6xz + 15xy - 25y^2 + 10yz + 6xz - 10yz + 4z^2 \]

4. **Combine like terms:**
- The \(xy\) terms:
\[ -15xy + 15xy = 0 \]
- The \(xz\) terms:
\[ 6xz + 6xz = 12xz \]
- The \(yz\) terms:
\[ 10yz - 10yz = 0 \]

5. **Final expression:**
\[ 9x^2 - 25y^2 + 12xz + 4z^2 \]

### Final Result:
The expanded form of \((3x + 5y + 2z)(3x - 5y + 2z)\) is:
\[ 9x^2 - 25y^2 + 12xz + 4z^2 \]