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

Answer

To expand the expression \((a - 2b - 5c)^2\), we will use the formula for the square of a trinomial. The formula states that:

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

In our case, we can rewrite \(a - 2b - 5c\) as \(a + (-2b) + (-5c)\). Now, we can identify \(b = -2b\) and \(c = -5c\).

### Step 1: Identify the components
Let:
- \(x = a\)
- \(y = -2b\)
- \(z = -5c\)

### Step 2: Apply the formula
Using the formula, we have:

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

### Step 3: Calculate each term
1. **Calculate \(x^2\)**:
\[ x^2 = a^2 \]

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

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

4. **Calculate \(2(xy + xz + yz)\)**:
- \(xy = a \cdot (-2b) = -2ab\)
- \(xz = a \cdot (-5c) = -5ac\)
- \(yz = (-2b) \cdot (-5c) = 10bc\)

Therefore,
\[ xy + xz + yz = -2ab - 5ac + 10bc \]
and
\[ 2(xy + xz + yz) = 2(-2ab - 5ac + 10bc) = -4ab - 10ac + 20bc \]

### Step 4: Combine all parts
Putting it all together, we have:

\[ (a - 2b - 5c)^2 = a^2 + 4b^2 + 25c^2 - 4ab + 20bc - 10ac \]

### Final Answer:
\[ (a - 2b - 5c)^2 = a^2 + 4b^2 + 25c^2 - 4ab + 20bc - 10ac \]

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