Suppose \(x^2+px+q\) can be factorised over integers as \((x+m)(x+n)\). If \(q<0\) and \(p>0\), which statement about \(m\) and \(n\) must be true?

Correct answer: They have opposite signs, and the positive number has greater magnitude.

The direct answer is A. Expanding \((x+m)(x+n)\) gives \(x^2+(m+n)x+mn\). Comparing coefficients with \(x^2+px+q\), we get \(m+n=p\) and \(mn=q\). Since \(q<0\), the product \(mn\) is negative, so one of \(m,n\) is positive and the other is negative. Since \(p>0\), their sum is positive. When two numbers have opposite signs, their sum is positive only when the positive number has the greater magnitude. Therefore A must be true. B and D are impossible because numbers with the same sign have a positive product or, if both negative, a negative sum. C would produce a negative sum, not the given positive \(p\). Memory cue: negative product means opposite signs; the sign of the sum tells which magnitude is larger.