Factorise:
`x ^(2) - 4x -12`

Answer

To factorise the expression \( x^2 - 4x - 12 \), we will follow these steps:

### Step 1: Identify the coefficients
The expression is in the standard quadratic form \( ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -4 \) (coefficient of \( x \))
- \( c = -12 \) (constant term)

### Step 2: Multiply \( a \) and \( c \)
We need to find two numbers that multiply to \( ac \) (which is \( 1 \times -12 = -12 \)) and add up to \( b \) (which is \( -4 \)).

### Step 3: Find the two numbers
The two numbers that satisfy these conditions are \( -6 \) and \( 2 \) because:
- \( -6 \times 2 = -12 \)
- \( -6 + 2 = -4 \)

### Step 4: Rewrite the middle term
We can rewrite the expression \( x^2 - 4x - 12 \) using the two numbers we found:
\[ x^2 - 6x + 2x - 12 \]

### Step 5: Group the terms
Now, we will group the terms:
\[ (x^2 - 6x) + (2x - 12) \]

### Step 6: Factor out the common terms
Now, we factor out the common terms from each group:
\[ x(x - 6) + 2(x - 6) \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \( (x - 6) \):
\[ (x - 6)(x + 2) \]

### Final Answer
Thus, the factorised form of \( x^2 - 4x - 12 \) is:
\[ (x - 6)(x + 2) \]

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