Factorise :
`3x^(2) - 11x - 4`

Answer

To factorise the expression \(3x^2 - 11x - 4\), we will use the method of splitting the middle term. Here are the steps:

### Step 1: Identify the coefficients
The given quadratic expression is:
\[ 3x^2 - 11x - 4 \]
Here, the coefficient of \(x^2\) (a) is 3, the coefficient of \(x\) (b) is -11, and the constant term (c) is -4.

### Step 2: Calculate the product of \(a\) and \(c\)
We need to find two numbers that multiply to \(a \cdot c\) (which is \(3 \cdot -4 = -12\)) and add up to \(b\) (which is -11).

### Step 3: Find the two numbers
The two numbers that satisfy these conditions are -12 and 1, since:
\[ -12 \times 1 = -12 \quad \text{and} \quad -12 + 1 = -11 \]

### Step 4: Rewrite the middle term
Now, we can rewrite the expression by splitting the middle term using the two numbers we found:
\[ 3x^2 - 12x + 1x - 4 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (3x^2 - 12x) + (1x - 4) \]

### Step 6: Factor out the common factors
Now, we factor out the common factors from each group:
\[ 3x(x - 4) + 1(x - 4) \]

### Step 7: Factor by grouping
Now, we can factor out the common binomial factor \((x - 4)\):
\[ (3x + 1)(x - 4) \]

### Final Answer
Thus, the factorised form of \(3x^2 - 11x - 4\) is:
\[ (3x + 1)(x - 4) \]

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