Factorise :
`6 + 11x + 3x^(2)`

Answer

To factorise the expression \(6 + 11x + 3x^2\), we will follow these steps:

### Step 1: Rearrange the expression
First, we can rearrange the expression in standard form, which is typically written with the highest power of \(x\) first. Thus, we rewrite it as:
\[ 3x^2 + 11x + 6 \]

### Step 2: Identify coefficients
In the quadratic expression \(3x^2 + 11x + 6\), we identify:
- \(a = 3\) (coefficient of \(x^2\))
- \(b = 11\) (coefficient of \(x\))
- \(c = 6\) (constant term)

### Step 3: Calculate the product of \(a\) and \(c\)
Next, we calculate the product of \(a\) and \(c\):
\[ a \cdot c = 3 \cdot 6 = 18 \]

### Step 4: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(18\) (the product we just calculated) and add up to \(11\) (the coefficient \(b\)). The numbers that satisfy this condition are \(9\) and \(2\) because:
\[ 9 \cdot 2 = 18 \quad \text{and} \quad 9 + 2 = 11 \]

### Step 5: Split the middle term
Now, we can split the middle term \(11x\) into \(9x + 2x\):
\[ 3x^2 + 9x + 2x + 6 \]

### Step 6: Group the terms
Next, we group the terms into two pairs:
\[ (3x^2 + 9x) + (2x + 6) \]

### Step 7: Factor out the common factors from each group
Now, we factor out the common factors from each group:
\[ 3x(x + 3) + 2(x + 3) \]

### Step 8: Factor out the common binomial factor
We can see that \((x + 3)\) is a common factor:
\[ (3x + 2)(x + 3) \]

### Final Answer
Thus, the factorised form of the expression \(6 + 11x + 3x^2\) is:
\[ (3x + 2)(x + 3) \]

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