Factorise:
`3x ^(2) - 4x -4`

Answer

To factorise the quadratic expression \(3x^2 - 4x - 4\), we can follow these steps:

### Step 1: Identify the coefficients
The quadratic expression is in the form \(ax^2 + bx + c\), where:
- \(a = 3\)
- \(b = -4\)
- \(c = -4\)

### Step 2: Calculate the product \(ac\)
We need to calculate the product of \(a\) and \(c\):
\[ ac = 3 \times (-4) = -12 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(-12\) (the value of \(ac\)) and add to \(-4\) (the value of \(b\)). The two numbers that satisfy this condition are \(-6\) and \(2\):
\[ -6 \times 2 = -12 \quad \text{and} \quad -6 + 2 = -4 \]

### Step 4: Rewrite the middle term
We can rewrite the quadratic expression by splitting the middle term using the two numbers found:
\[ 3x^2 - 6x + 2x - 4 \]

### Step 5: Group the terms
Now, we will group the terms:
\[ (3x^2 - 6x) + (2x - 4) \]

### Step 6: Factor out the common factors from each group
From the first group \(3x^2 - 6x\), we can factor out \(3x\):
\[ 3x(x - 2) \]
From the second group \(2x - 4\), we can factor out \(2\):
\[ 2(x - 2) \]

### Step 7: Combine the factored groups
Now we can combine the factored groups:
\[ 3x(x - 2) + 2(x - 2) \]
We can factor out the common binomial \((x - 2)\):
\[ (x - 2)(3x + 2) \]

### Final Result
Thus, the factorised form of the expression \(3x^2 - 4x - 4\) is:
\[ \boxed{(x - 2)(3x + 2)} \]

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