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

Answer

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

### Step 1: Identify the coefficients
The given quadratic expression is in the form \(ax^2 + bx + c\), where:
- \(a = 6\)
- \(b = 11\)
- \(c = 3\)

### Step 2: Multiply \(a\) and \(c\)
Calculate the product of \(a\) and \(c\):
\[ a \cdot c = 6 \cdot 3 = 18 \]

### Step 3: Find two numbers that add to \(b\) and multiply to \(ac\)
We need to find two numbers that add up to \(b = 11\) and multiply to \(ac = 18\). The numbers that satisfy this condition are \(9\) and \(2\) because:
\[ 9 + 2 = 11 \quad \text{and} \quad 9 \cdot 2 = 18 \]

### Step 4: Rewrite the middle term
Now we can rewrite the expression by splitting the middle term \(11x\) into \(9x + 2x\):
\[ 6x^2 + 9x + 2x + 3 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (6x^2 + 9x) + (2x + 3) \]

### Step 6: Factor out the common factors
Now, we factor out the common factors from each group:
- From the first group \(6x^2 + 9x\), we can factor out \(3x\):
\[ 3x(2x + 3) \]
- From the second group \(2x + 3\), we can factor out \(1\):
\[ 1(2x + 3) \]

### Step 7: Combine the factors
Now we can combine the factored terms:
\[ 3x(2x + 3) + 1(2x + 3) = (2x + 3)(3x + 1) \]

### Final Answer
Thus, the factorised form of \(6x^2 + 11x + 3\) is:
\[ (2x + 3)(3x + 1) \]

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