Factorise:
`7x ^(2) - 19 x -6`

Answer

To factorise the expression \( 7x^2 - 19x - 6 \), we will follow these steps:

### Step 1: Identify coefficients
The expression is \( 7x^2 - 19x - 6 \).
- Coefficient of \( x^2 \) (let's call it \( a \)) = 7
- Coefficient of \( x \) (let's call it \( b \)) = -19
- Constant term (let's call it \( c \)) = -6

### Step 2: Multiply \( a \) and \( c \)
We need to multiply the coefficient of \( x^2 \) (which is 7) by the constant term (which is -6):
\[ a \cdot c = 7 \cdot (-6) = -42 \]

### Step 3: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to -42 and add up to -19.
The two numbers that satisfy this are -21 and 2, since:
\[ -21 \cdot 2 = -42 \quad \text{and} \quad -21 + 2 = -19 \]

### Step 4: Rewrite the middle term
Now, we can rewrite the expression \( 7x^2 - 19x - 6 \) using the two numbers we found:
\[ 7x^2 - 21x + 2x - 6 \]

### Step 5: Group the terms
Next, we will group the terms:
\[ (7x^2 - 21x) + (2x - 6) \]

### Step 6: Factor out the common factors from each group
Now, we factor out the common factors from each group:
- From \( 7x^2 - 21x \), we can factor out \( 7x \):
\[ 7x(x - 3) \]

- From \( 2x - 6 \), we can factor out \( 2 \):
\[ 2(x - 3) \]

### Step 7: Combine the factors
Now we can combine the factored groups:
\[ 7x(x - 3) + 2(x - 3) \]
We can see that \( (x - 3) \) is a common factor:
\[ (x - 3)(7x + 2) \]

### Final Answer
Thus, the factorised form of \( 7x^2 - 19x - 6 \) is:
\[ (7x + 2)(x - 3) \]

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