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

Answer

To expand the expression \((3x - 2y + 5z)^2\), we can use the formula for the square of a trinomial, which is given by:

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

In our case, we can identify:
- \(a = 3x\)
- \(b = -2y\)
- \(c = 5z\)

Now, we will apply the formula step by step.

### Step 1: Calculate \(a^2\)
\[ a^2 = (3x)^2 = 9x^2 \]

### Step 2: Calculate \(b^2\)
\[ b^2 = (-2y)^2 = 4y^2 \]

### Step 3: Calculate \(c^2\)
\[ c^2 = (5z)^2 = 25z^2 \]

### Step 4: Calculate \(2ab\)
\[ 2ab = 2 \cdot (3x) \cdot (-2y) = -12xy \]

### Step 5: Calculate \(2bc\)
\[ 2bc = 2 \cdot (-2y) \cdot (5z) = -20yz \]

### Step 6: Calculate \(2ca\)
\[ 2ca = 2 \cdot (5z) \cdot (3x) = 30xz \]

### Step 7: Combine all the results
Now we can combine all the parts we calculated:
\[ (3x - 2y + 5z)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \]
\[ = 9x^2 + 4y^2 + 25z^2 - 12xy - 20yz + 30xz \]

Thus, the expanded form of \((3x - 2y + 5z)^2\) is:
\[ 9x^2 + 4y^2 + 25z^2 - 12xy - 20yz + 30xz \]