If the discriminant of a quadratic equation is \(D=(z+1)^2-9\), in which interval must \(z\) lie for the equation to have no real roots?

Correct answer: \(-4<z<2\)

A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Thus, \((z+1)^2-9<0\), which gives \((z+1)^2<9\). Therefore, \(-3<z+1<3\), and hence \(-4<z<2\). So, option A is correct. Remember that \(D=0\) gives two equal real roots, so the endpoints \(-4\) and \(2\) are excluded from the interval.