What are the factors of \(x^2 + 5x − 24\)?

Correct answer: (x + 8)(x − 3)

The direct answer is A. Because the coefficient of \(x^2\) is \(1\), look for two integers whose product is the constant term \(−24\) and whose sum is the middle coefficient \(5\). The pair \(8\) and \(−3\) works: \(8×(−3)=−24\) and \(8+(−3)=5\). Therefore \(x^2+5x−24=(x+8)(x−3)\). Expanding confirms it: \(x^2−3x+8x−24=x^2+5x−24\). B uses \(−8\) and \(3\), whose sum is \(−5\), so it gives \(x^2−5x−24\). C has product \(−24\) but sum \(2\), producing the wrong middle term. D has product \(−24\) but sum \(10\), also incorrect. When the constant is negative, the two numbers have opposite signs; the positive number must have the larger absolute value here because the middle coefficient is positive. Memory cue: for \(x^2+bx+c\), find \(p\) and \(q\) with \(pq=c\) and \(p+q=b\), then write \((x+p)(x+q)\).