Which statement correctly describes the nature of the roots of \(x^2+2px+p^2+1=0\)?

Correct answer: There are no real roots for any real \(p\)

Here, \(a=1\), \(b=2p\), and \(c=p^2+1\). Thus, the discriminant is \(D=b^2-4ac=(2p)^2-4(1)(p^2+1)=-4\), which remains negative for every real \(p\). Therefore, the equation has no real roots. Option D is also incorrect because putting \(p=0\) gives \(x^2+1=0\), which has no real roots. Exam tip: For a quadratic equation, \(D<0\) means the roots are non-real conjugates.