Let one root of the equation $x^2 + \ell x + m = 0$ is square of other root. If $m \in R$ then

Let roots be a and a^2. Then a + a^2 = -l and a^3 = m. For real roots, the discriminant D = l^2 - 4m >= 0. Substituting m = a^3 and l = -(a + a^2), we get (a + a^2)^2 - 4a^3 >= 0, which simplifies to a^2(a-1)^2 >= 0. This is always true for real a. The condition for the roots to be real is l^2 >= 4m. Analysis of the range of l leads to the given set.