If the trinomial \(x^2+mx+n\) is factorised as \((x+p)(x+q)\), which relation between \(p\) and \(q\) is correct?

Correct answer: \(p+q=m,\ pq=n\)

The direct answer is A. Expand the proposed factors carefully: \((x+p)(x+q)=x^2+qx+px+pq=x^2+(p+q)x+pq\). Now compare this with \(x^2+mx+n\). The coefficient of \(x\) must be the same, so \(p+q=m\). The constant term must also be the same, so \(pq=n\). Option A states exactly these two relations. Option B interchanges the roles of the middle coefficient and the constant term; addition of the numbers belongs to the coefficient of \(x\), while multiplication belongs to the constant term. Option C uses a difference, but expansion produces a sum \(p+q\), not generally \(p-q\). Option D incorrectly associates a difference with the constant term. The safest method is always to multiply the two binomials and compare like terms. Memory cue: “sum makes the middle term; product makes the last term.”