Factorise:
`x ^(2) + x - 132 `

Answer

To factorise the expression \( x^2 + x - 132 \), we will follow these steps:

### Step 1: Identify the coefficients
The expression is in the form \( ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 1 \) (coefficient of \( x \))
- \( c = -132 \) (constant term)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( a \cdot c = 1 \cdot (-132) = -132 \) and add up to \( b = 1 \).

### Step 3: List the factor pairs of \(-132\)
The factor pairs of \(-132\) that we can consider are:
- \( (1, -132) \)
- \( (-1, 132) \)
- \( (2, -66) \)
- \( (-2, 66) \)
- \( (3, -44) \)
- \( (-3, 44) \)
- \( (4, -33) \)
- \( (-4, 33) \)
- \( (6, -22) \)
- \( (-6, 22) \)
- \( (11, -12) \)
- \( (-11, 12) \)

### Step 4: Identify the correct pair
Among these pairs, we need to find the pair that adds up to \( 1 \). The pair \( (12, -11) \) works because:
- \( 12 + (-11) = 1 \)
- \( 12 \times (-11) = -132 \)

### Step 5: Rewrite the middle term
Now, we can rewrite the expression \( x^2 + x - 132 \) using the numbers we found:
\[ x^2 + 12x - 11x - 132 \]

### Step 6: Group the terms
Next, we group the terms:
\[ (x^2 + 12x) + (-11x - 132) \]

### Step 7: Factor by grouping
Now we factor out the common factors from each group:
\[ x(x + 12) - 11(x + 12) \]

### Step 8: Factor out the common binomial
Now we can factor out the common binomial \( (x + 12) \):
\[ (x - 11)(x + 12) \]

### Final Answer
Thus, the factorised form of \( x^2 + x - 132 \) is:
\[ (x - 11)(x + 12) \]

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