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The expression given is \( 3x^2 - 7xy - 6y^2 \). 

To factorise this quadratic expression in terms of \( x \) and \( y \), we will apply the method of factorisation by grouping.

### Step 1: Multiply the coefficient of \( x^2 \) (which is 3) and the constant term (coefficient of \( y^2 \), which is -6). The result is \( 3 \times -6 = -18 \).

### Step 2: Find two numbers that multiply to give -18 and add to give the middle coefficient, -7 (the coefficient of \( xy \)).
The numbers are -9 and 2 because:
\[
-9 \times 2 = -18 \quad \text{and} \quad -9 + 2 = -7.
\]

### Step 3: Rewrite the middle term using these two numbers:
\[
3x^2 - 9xy + 2xy - 6y^2.
\]

### Step 4: Group the terms in pairs:
\[
(3x^2 - 9xy) + (2xy - 6y^2).
\]

### Step 5: Factor out the common factors from each group:
\[
3x(x - 3y) + 2y(x - 3y).
\]

### Step 6: Now, factor out the common binomial factor \( (x - 3y) \):
\[
(3x + 2y)(x - 3y).
\]

Thus, the factorised form of the expression \( 3x^2 - 7xy - 6y^2 \) is:
\[
(3x + 2y)(x - 3y).
\]

Would you like further details or clarification on any step?

### Related questions:
1. What is the method of factorisation by grouping?
2. How do you identify common factors in each term?
3. Can all quadratic expressions be factorised this way?
4. What happens if the quadratic cannot be factorised into rational terms?
5. How does this factorisation apply to solving quadratic equations?

### Tip:
Always verify your factorisation by expanding the factors to ensure the original expression is recovered.

Learn how to factorise the quadratic expression 3x^2 - 7xy - 6y^2 using the method of grouping. This step-by-step explanation covers key algebraic concepts and formulas, suitable for students in Grades 9-12.

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