Simplify: (2p - q)$^2$ - (p + q)$^2$

Analyze the question: We need to expand both squares, then subtract and factorise the result.

Concept: Use the expansion (a ± b)² = a² ± 2ab + b². You can also use the difference of two squares: a² - b² = (a - b)(a + b).

Method 1: Expand

(2p - q)² = 4p² - 4pq + q²

(p + q)² = p² + 2pq + q²

Subtract:

(4p² - 4pq + q²) - (p² + 2pq + q²) = 4p² - 4pq + q² - p² - 2pq - q² = 3p² - 6pq

Factorise:

= 3p(p - 2q)

Method 2: Difference of two squares (a check)

Let a = 2p - q and b = p + q.

• a - b = (2p - q) - (p + q) = p - 2q
• a + b = (2p - q) + (p + q) = 3p

So the product is (p - 2q)(3p) = 3p(p - 2q). ✓

Answer: Option A, 3p(p - 2q).

Tip: When you see a square minus a square, try the difference of two squares first. It is quicker and gives you a factorised answer straight away, so you avoid sign errors when subtracting a bracket. Always remember that the minus sign changes the sign of every term in the second bracket.