If $\alpha, \beta, \gamma$ are the roots of the equation $x^{3} + px^{2} + qx + r = 0$ then the coefficient of $x$ in the cubic equation whose roots are $\alpha (\beta + \gamma), \beta (\gamma + \alpha)$ and $\gamma (\alpha + \beta)$ is

Roots are a, b, c. a+b+c = -p, ab+bc+ca = q, abc = -r. New roots: a(b+c), b(c+a), c(a+b). Note a(b+c) = a(-p-a) = -ap - a^2. This is getting complicated. Let's use the property that the sum of new roots is -2(ab+bc+ca) = -2q. The sum of products taken two at a time is the coefficient of x. The calculation leads to q^2 + pr.