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

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

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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.
$$ x^2 - x - 6 = (x + 2)(x - 3) $$

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

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