Factorise the following : (i) `2x^(2)+9x+9` (ii) `2x^(2)+9x-5`

Answer

Let's factorize the given expressions step by step.

### (i) Factorizing \(2x^2 + 9x + 9\)

1. **Identify coefficients**:
- Here, \(a = 2\), \(b = 9\), and \(c = 9\).

2. **Calculate \(ab\)**:
- We need to find two numbers that add up to \(b\) (which is 9) and multiply to \(ac\) (which is \(2 \times 9 = 18\)).
- So, we need two numbers \(m\) and \(n\) such that:
- \(m + n = 9\)
- \(m \times n = 18\)

3. **Find suitable pairs**:
- The pairs that satisfy these conditions are \(3\) and \(6\) because:
- \(3 + 6 = 9\)
- \(3 \times 6 = 18\)

4. **Rewrite the expression**:
- Rewrite \(9x\) as \(3x + 6x\):
\[ 2x^2 + 3x + 6x + 9 \]

5. **Group the terms**:
- Group the first two and the last two terms:
\[ (2x^2 + 3x) + (6x + 9) \]

6. **Factor out common terms**:
- From the first group \(2x^2 + 3x\), factor out \(x\):
\[ x(2x + 3) \]
- From the second group \(6x + 9\), factor out \(3\):
\[ 3(2x + 3) \]

7. **Combine the factors**:
- Now, we can combine the two groups:
\[ (x + 3)(2x + 3) \]

### Final Factorization for (i):
\[ 2x^2 + 9x + 9 = (x + 3)(2x + 3) \]

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### (ii) Factorizing \(2x^2 + 9x - 5\)

1. **Identify coefficients**:
- Here, \(a = 2\), \(b = 9\), and \(c = -5\).

2. **Calculate \(ab\)**:
- We need to find two numbers that add up to \(b\) (which is 9) and multiply to \(ac\) (which is \(2 \times -5 = -10\)).
- So, we need two numbers \(m\) and \(n\) such that:
- \(m + n = 9\)
- \(m \times n = -10\)

3. **Find suitable pairs**:
- The pairs that satisfy these conditions are \(10\) and \(-1\) because:
- \(10 - 1 = 9\)
- \(10 \times -1 = -10\)

4. **Rewrite the expression**:
- Rewrite \(9x\) as \(10x - 1x\):
\[ 2x^2 + 10x - x - 5 \]

5. **Group the terms**:
- Group the first two and the last two terms:
\[ (2x^2 + 10x) + (-x - 5) \]

6. **Factor out common terms**:
- From the first group \(2x^2 + 10x\), factor out \(2x\):
\[ 2x(x + 5) \]
- From the second group \(-x - 5\), factor out \(-1\):
\[ -1(x + 5) \]

7. **Combine the factors**:
- Now, we can combine the two groups:
\[ (2x - 1)(x + 5) \]

### Final Factorization for (ii):
\[ 2x^2 + 9x - 5 = (2x - 1)(x + 5) \]

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