Use the same technique as in (4 and 5) above to factorise the following. Show all the Steps. \( \begin{array}{l}7.1 \\ x^{2}-x-6 x^{2}-x-6 \\ \text { Diagram: }\end{array} \)

To factor the quadratic expression \( x^2 - x - 6 \), we will follow these steps: 1. **Identify the coefficients**: The quadratic expression can be written in the standard form \( ax^2 + bx + c \), where: - \( a = 1 \) (coefficient of \( x^2 \)) - \( b = -1 \) (coefficient of \( x \)) - \( c = -6 \) (constant term) 2. **Find two numbers that multiply to \( ac \) and add to \( b \)**: - Here, \( ac = 1 \times (-6) = -6 \). - We need to find two numbers that multiply to \(-6\) and add to \(-1\). The pairs of factors of \(-6\) are: - \( (1, -6) \) - \( (-1, 6) \) - \( (2, -3) \) - \( (-2, 3) \) The pair that adds up to \(-1\) is \( (2, -3) \). 3. **Rewrite the middle term using the two numbers**: We can rewrite the expression \( x^2 - x - 6 \) as: \[ x^2 + 2x - 3x - 6 \] 4. **Group the terms**: Now, we group the terms: \[ (x^2 + 2x) + (-3x - 6) \] 5. **Factor by grouping**: - From the first group \( (x^2 + 2x) \), we can factor out \( x \): \[ x(x + 2) \] - From the second group \( (-3x - 6) \), we can factor out \(-3\): \[ -3(x + 2) \] Now we have: \[ x(x + 2) - 3(x + 2) \] 6. **Factor out the common binomial**: We can factor out the common factor \( (x + 2) \): \[ (x + 2)(x - 3) \] Thus, the factored form of the expression \( x^2 - x - 6 \) is: \[ (x + 2)(x - 3) \] This completes the factorization process.