Expand
(i) `(a+2b+5c)^2`
(ii) ` (2a+b+c)^2`
(iii) ` (a-2b-3c)^2`

Answer

Let's expand the given expressions step by step.

### (i) Expand `(a + 2b + 5c)^2`

1. **Identify the terms**: Here, we have \( a \), \( 2b \), and \( 5c \).
2. **Use the formula**: The expansion of \( (x + y + z)^2 \) is given by \( x^2 + y^2 + z^2 + 2xy + 2yz + 2zx \).
3. **Apply the formula**:
- \( x = a \)
- \( y = 2b \)
- \( z = 5c \)
- Calculate \( x^2 = a^2 \)
- Calculate \( y^2 = (2b)^2 = 4b^2 \)
- Calculate \( z^2 = (5c)^2 = 25c^2 \)
- Calculate \( 2xy = 2(a)(2b) = 4ab \)
- Calculate \( 2yz = 2(2b)(5c) = 20bc \)
- Calculate \( 2zx = 2(a)(5c) = 10ac \)
4. **Combine all the terms**:
\[ (a + 2b + 5c)^2 = a^2 + 4b^2 + 25c^2 + 4ab + 20bc + 10ac \]

### (ii) Expand `(2a + b + c)^2`

1. **Identify the terms**: Here, we have \( 2a \), \( b \), and \( c \).
2. **Use the formula**: Again, we will use the same expansion formula.
3. **Apply the formula**:
- \( x = 2a \)
- \( y = b \)
- \( z = c \)
- Calculate \( x^2 = (2a)^2 = 4a^2 \)
- Calculate \( y^2 = b^2 \)
- Calculate \( z^2 = c^2 \)
- Calculate \( 2xy = 2(2a)(b) = 4ab \)
- Calculate \( 2yz = 2(b)(c) = 2bc \)
- Calculate \( 2zx = 2(2a)(c) = 4ac \)
4. **Combine all the terms**:
\[ (2a + b + c)^2 = 4a^2 + b^2 + c^2 + 4ab + 2bc + 4ac \]

### (iii) Expand `(a - 2b - 3c)^2`

1. **Identify the terms**: Here, we have \( a \), \( -2b \), and \( -3c \).
2. **Use the formula**: We will use the same expansion formula.
3. **Apply the formula**:
- \( x = a \)
- \( y = -2b \)
- \( z = -3c \)
- Calculate \( x^2 = a^2 \)
- Calculate \( y^2 = (-2b)^2 = 4b^2 \)
- Calculate \( z^2 = (-3c)^2 = 9c^2 \)
- Calculate \( 2xy = 2(a)(-2b) = -4ab \)
- Calculate \( 2yz = 2(-2b)(-3c) = 12bc \)
- Calculate \( 2zx = 2(a)(-3c) = -6ac \)
4. **Combine all the terms**:
\[ (a - 2b - 3c)^2 = a^2 + 4b^2 + 9c^2 - 4ab + 12bc - 6ac \]

### Summary of Results:
1. \( (a + 2b + 5c)^2 = a^2 + 4b^2 + 25c^2 + 4ab + 20bc + 10ac \)
2. \( (2a + b + c)^2 = 4a^2 + b^2 + c^2 + 4ab + 2bc + 4ac \)
3. \( (a - 2b - 3c)^2 = a^2 + 4b^2 + 9c^2 - 4ab + 12bc - 6ac \)