What are the factors of 2x$^2$ - xy - 3y$^2$ ?

Analyze the question: We need to factorise the quadratic expression 2x² − xy − 3y².

Concept: For a quadratic of the form ax² + bxy + cy², find two numbers whose product is a×c and whose sum is b. Then split the middle term and factorise by grouping.

Solution:

• a × c = 2 × (−3) = −6
• b = −1
• The two numbers are −3 and +2, since (−3)(2) = −6 and −3 + 2 = −1.

Split the middle term:

2x² − 3xy + 2xy − 3y²

Group:

x(2x − 3y) + y(2x − 3y)

= (2x − 3y)(x + y)

Check by expanding: (2x − 3y)(x + y) = 2x² + 2xy − 3xy − 3y² = 2x² − xy − 3y² ✓

Answer: Option D, (2x − 3y)(x + y).

Tip: Quickly check the signs by expanding. The product of the last terms must give −3y², so the signs in the brackets must differ. Then test which arrangement gives the middle term −xy. Option C, (2x + 3y)(x − y), gives +xy instead, so it fails the check.