Factorise:
`6x ^(2) - 17x -3`

Answer

To factorise the expression \( 6x^2 - 17x - 3 \), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is \( 6x^2 - 17x - 3 \).
- Coefficient of \( x^2 \) (a) = 6
- Coefficient of \( x \) (b) = -17
- Constant term (c) = -3

### Step 2: Multiply \( a \) and \( c \)
We need to multiply the coefficient of \( x^2 \) (which is 6) with the constant term (which is -3):
\[ a \cdot c = 6 \cdot (-3) = -18 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to -18 (the result from Step 2) and add up to -17 (the coefficient of \( x \)):
The numbers are -18 and 1, because:
\[ -18 \cdot 1 = -18 \quad \text{and} \quad -18 + 1 = -17 \]

### Step 4: Rewrite the middle term using the two numbers
We can rewrite the expression \( 6x^2 - 17x - 3 \) by splitting the middle term (-17x) into -18x and +1x:
\[ 6x^2 - 18x + 1x - 3 \]

### Step 5: Factor by grouping
Now, we group the terms:
\[ (6x^2 - 18x) + (1x - 3) \]
Now, we factor out the common factors from each group:
- From the first group \( 6x^2 - 18x \), we can factor out \( 6x \):
\[ 6x(x - 3) \]
- From the second group \( 1x - 3 \), we can factor out \( 1 \):
\[ 1(x - 3) \]

### Step 6: Combine the factors
Now we can combine the factored groups:
\[ 6x(x - 3) + 1(x - 3) = (6x + 1)(x - 3) \]

### Final Answer
Thus, the factorised form of \( 6x^2 - 17x - 3 \) is:
\[ (6x + 1)(x - 3) \]

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