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

Answer

To expand the expression \((5a - 3b + c)^2\), we will use the formula for the square of a trinomial, which is:

\[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx \]

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

Now, we will apply the formula step by step.

### Step 1: Calculate \(x^2\)
\[ x^2 = (5a)^2 = 25a^2 \]

### Step 2: Calculate \(y^2\)
\[ y^2 = (-3b)^2 = 9b^2 \]

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

### Step 4: Calculate \(2xy\)
\[ 2xy = 2 \cdot (5a) \cdot (-3b) = -30ab \]

### Step 5: Calculate \(2yz\)
\[ 2yz = 2 \cdot (-3b) \cdot c = -6bc \]

### Step 6: Calculate \(2zx\)
\[ 2zx = 2 \cdot c \cdot (5a) = 10ac \]

### Step 7: Combine all the results
Now, we can combine all the calculated parts:
\[ (5a - 3b + c)^2 = 25a^2 + 9b^2 + c^2 - 30ab - 6bc + 10ac \]

Thus, the expanded form of \((5a - 3b + c)^2\) is:
\[ 25a^2 + 9b^2 + c^2 - 30ab - 6bc + 10ac \]