Expand :
`(i) (a+b-c)^(2) " " (ii) (a-2b-5c)^(2) " " (iii) (3a-2b-5c)^(2) " " (iv) (2x+(1)/(x)+1)^(2)`.

Answer

Let's expand the given expressions step by step.

### (i) Expand \( (a + b - c)^2 \)

1. **Identify the terms**: Here, \( a \) is \( a \), \( b \) is \( b \), and \( c \) is \( -c \).
2. **Apply the formula**:
\[ (a + b - c)^2 = a^2 + b^2 + (-c)^2 + 2ab + 2b(-c) + 2(-c)a \]
3. **Simplify**:
\[ = a^2 + b^2 + c^2 + 2ab - 2bc - 2ca \]

### (ii) Expand \( (a - 2b - 5c)^2 \)

1. **Identify the terms**: Here, \( a \) is \( a \), \( b \) is \( -2b \), and \( c \) is \( -5c \).
2. **Apply the formula**:
\[ (a - 2b - 5c)^2 = a^2 + (-2b)^2 + (-5c)^2 + 2a(-2b) + 2(-2b)(-5c) + 2(-5c)a \]
3. **Simplify**:
\[ = a^2 + 4b^2 + 25c^2 - 4ab + 20bc - 10ca \]

### (iii) Expand \( (3a - 2b - 5c)^2 \)

1. **Identify the terms**: Here, \( a \) is \( 3a \), \( b \) is \( -2b \), and \( c \) is \( -5c \).
2. **Apply the formula**:
\[ (3a - 2b - 5c)^2 = (3a)^2 + (-2b)^2 + (-5c)^2 + 2(3a)(-2b) + 2(-2b)(-5c) + 2(-5c)(3a) \]
3. **Simplify**:
\[ = 9a^2 + 4b^2 + 25c^2 - 12ab + 20bc - 30ca \]

### (iv) Expand \( (2x + \frac{1}{x} + 1)^2 \)

1. **Identify the terms**: Here, \( a \) is \( 2x \), \( b \) is \( \frac{1}{x} \), and \( c \) is \( 1 \).
2. **Apply the formula**:
\[ (2x + \frac{1}{x} + 1)^2 = (2x)^2 + \left(\frac{1}{x}\right)^2 + 1^2 + 2(2x)(\frac{1}{x}) + 2(\frac{1}{x})(1) + 2(1)(2x) \]
3. **Simplify**:
\[ = 4x^2 + \frac{1}{x^2} + 1 + 4 + \frac{2}{x} + 4x \]
\[ = 4x^2 + \frac{1}{x^2} + 5 + 4x + \frac{2}{x} \]

### Final Answers:
1. \( (a + b - c)^2 = a^2 + b^2 + c^2 + 2ab - 2bc - 2ca \)
2. \( (a - 2b - 5c)^2 = a^2 + 4b^2 + 25c^2 - 4ab + 20bc - 10ca \)
3. \( (3a - 2b - 5c)^2 = 9a^2 + 4b^2 + 25c^2 - 12ab + 20bc - 30ca \)
4. \( (2x + \frac{1}{x} + 1)^2 = 4x^2 + \frac{1}{x^2} + 5 + 4x + \frac{2}{x} \)