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

Answer

To expand the expression \((a + 2b - 5c)^2\), we will 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 have \(a\), \(b = 2b\), and \(c = -5c\). Let's expand step by step.

### Step 1: Identify the terms
We can identify our terms as:
- \(x = a\)
- \(y = 2b\)
- \(z = -5c\)

### Step 2: Apply the formula
Using the formula, we will expand \((x + y + z)^2\):

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

### 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 \(2xy\)**:
\[ 2xy = 2 \cdot a \cdot (2b) = 4ab \]

5. **Calculate \(2yz\)**:
\[ 2yz = 2 \cdot (2b) \cdot (-5c) = -20bc \]

6. **Calculate \(2zx\)**:
\[ 2zx = 2 \cdot (-5c) \cdot a = -10ac \]

### Step 4: Combine all terms
Now, we combine all the calculated terms:

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

### Final Answer
Thus, the expanded form of \((a + 2b - 5c)^2\) is:

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

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