If the quadratic expression (x - 2)(x - p) expands to x^2 - a x + 6, what is the value of the difference (a - p)?

Introduction / Context:
This question checks whether you can match a factored quadratic expression with its expanded standard form and correctly compare coefficients. It is a typical algebra skill tested in school mathematics, aptitude exams and placement tests where you must relate factor form and coefficient form without fully solving the equation.

Given Data / Assumptions:
- We know (x - 2)(x - p) = x^2 - a x + 6.
- a and p are real constants.
- We must determine the value of (a - p).

Concept / Approach:
The method is to expand the left hand side and then compare coefficients of like powers of x with the expression on the right. This yields equations linking a and p. Once we know both a and p, we can compute a - p directly. No solving for x is required.

Step-by-Step Solution:
Expand the product (x - 2)(x - p): (x - 2)(x - p) = x^2 - (2 + p)x + 2p. We are told this equals x^2 - a x + 6. Compare coefficients of x^2: both sides have coefficient 1, so this gives no new information. Compare coefficients of x: we get -(2 + p) = -a, so a = 2 + p. Compare constant terms: 2p = 6, so p = 3. Substitute p = 3 into a = 2 + p to get a = 2 + 3 = 5. Therefore, a - p = 5 - 3 = 2.

Verification / Alternative check:
If p = 3, then the factorised form is (x - 2)(x - 3) which expands to x^2 - 5x + 6. This means a = 5 and the expression x^2 - a x + 6 is indeed x^2 - 5x + 6, confirming that a - p = 2 is consistent.

Why Other Options Are Wrong:
Options 0, 1 and 3 would correspond to incorrect relationships between a and p, usually arising from sign errors when matching -(2 + p) with -a. The value 4 is another distractor that does not satisfy both the x coefficient and constant term comparison at the same time.

Common Pitfalls:
Learners sometimes forget that the coefficient of x in the product (x - 2)(x - p) is the negative of the sum of the roots, so they may set a = 2 - p or similar, which is incorrect. Another pitfall is to treat 2p = 6 as p = 6 / 2 but then forget to update a accordingly. Careful comparison of coefficients avoids these mistakes.

Final Answer:
The required value of the difference is 2.