Factorize following expressions
(i) `x^(2)+3x-40" (ii)"x^2-3x-40" (iii)"x^(2)+5x-14`
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Answer

To factorize the given expressions, we will follow a systematic approach for each expression.

### (i) Factorizing \( x^2 + 3x - 40 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = 3 \), and \( c = -40 \).
2. **Find two numbers that multiply to \( ac \) (which is \( 1 \times -40 = -40 \)) and add to \( b \) (which is \( 3 \))**: The numbers are \( 8 \) and \( -5 \) because \( 8 \times -5 = -40 \) and \( 8 + (-5) = 3 \).
3. **Rewrite the middle term**: Rewrite \( 3x \) as \( 8x - 5x \):
\[ x^2 + 8x - 5x - 40 \]
4. **Group the terms**: Group the first two and the last two terms:
\[ (x^2 + 8x) + (-5x - 40) \]
5. **Factor out the common terms**:
\[ x(x + 8) - 5(x + 8) \]
6. **Factor by grouping**: Now factor out \( (x + 8) \):
\[ (x - 5)(x + 8) \]

### (ii) Factorizing \( x^2 - 3x - 40 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = -3 \), and \( c = -40 \).
2. **Find two numbers that multiply to \( ac \) (which is \( -40 \)) and add to \( b \) (which is \( -3 \))**: The numbers are \( -8 \) and \( 5 \) because \( -8 \times 5 = -40 \) and \( -8 + 5 = -3 \).
3. **Rewrite the middle term**: Rewrite \( -3x \) as \( -8x + 5x \):
\[ x^2 - 8x + 5x - 40 \]
4. **Group the terms**: Group the first two and the last two terms:
\[ (x^2 - 8x) + (5x - 40) \]
5. **Factor out the common terms**:
\[ x(x - 8) + 5(x - 8) \]
6. **Factor by grouping**: Now factor out \( (x - 8) \):
\[ (x + 5)(x - 8) \]

### (iii) Factorizing \( x^2 + 5x - 14 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = 5 \), and \( c = -14 \).
2. **Find two numbers that multiply to \( ac \) (which is \( -14 \)) and add to \( b \) (which is \( 5 \))**: The numbers are \( 7 \) and \( -2 \) because \( 7 \times -2 = -14 \) and \( 7 + (-2) = 5 \).
3. **Rewrite the middle term**: Rewrite \( 5x \) as \( 7x - 2x \):
\[ x^2 + 7x - 2x - 14 \]
4. **Group the terms**: Group the first two and the last two terms:
\[ (x^2 + 7x) + (-2x - 14) \]
5. **Factor out the common terms**:
\[ x(x + 7) - 2(x + 7) \]
6. **Factor by grouping**: Now factor out \( (x + 7) \):
\[ (x - 2)(x + 7) \]

### Summary of Factorized Forms
- (i) \( x^2 + 3x - 40 = (x - 5)(x + 8) \)
- (ii) \( x^2 - 3x - 40 = (x + 5)(x - 8) \)
- (iii) \( x^2 + 5x - 14 = (x - 2)(x + 7) \)