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

Answer

To expand the expression \((5x - 3y - 2)^2\), we will use the formula for the square of a trinomial, which states:

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

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

Now, we will apply the formula step by step.

### Step 1: Calculate \(a^2\), \(b^2\), and \(c^2\)

1. \(a^2 = (5x)^2 = 25x^2\)
2. \(b^2 = (-3y)^2 = 9y^2\)
3. \(c^2 = (-2)^2 = 4\)

### Step 2: Calculate the cross products \(2ab\), \(2bc\), and \(2ca\)

1. \(2ab = 2 \cdot (5x) \cdot (-3y) = -30xy\)
2. \(2bc = 2 \cdot (-3y) \cdot (-2) = 12y\)
3. \(2ca = 2 \cdot (-2) \cdot (5x) = -20x\)

### Step 3: Combine all the results

Now, we combine all the terms we calculated:

\[ (5x - 3y - 2)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \]

Substituting the values we found:

\[ = 25x^2 + 9y^2 + 4 - 30xy + 12y - 20x \]

### Final Result

Thus, the expanded form of \((5x - 3y - 2)^2\) is:

\[ 25x^2 + 9y^2 + 4 - 30xy + 12y - 20x \]

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