f(x), g(x), 2 quadratic polynomials:|f(x)|≥|g(x)|∀x∈R. Find the number of distinct roots of equation h(x)h″(x)+(h′(x))2=0 if h(x)=f(x)g(x)

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Let&amp;nbsp;f(x)=a(x−p)(x−q), where&amp;nbsp;a≠0,p,q∈R. Then&amp;nbsp;|g(p)|≤|f(p)|=0⇒g(p)=0, so&amp;nbsp;g(x)=b(x−p)(x−q)&amp;nbsp;for some real&amp;nbsp;b≠0. Thus we have&amp;nbsp;h(x)=c(x−p)2(x−q)2&amp;nbsp;for some real&amp;nbsp;c≠0.The condition&amp;nbsp;hh&amp;nbsp;″+(h′)2=0&amp;nbsp;is the same as&amp;nbsp;dd&amp;nbsp;dx&amp;nbsp;hh′=0.However, note h(x) has two double roots, hence shares those roots with h&#039;(x). The third root of the cubic h&#039;(x) must also then be real, between those two roots p,q. Thus in all, hh&#039; is a seventh degree polynomial with roots of multiplicity 3 at p,q, and one root at some r between p,q.This implies the derivative of hh&#039; must have all six roots real, two with multiplicity two at p,q, and one each between p,r and r,q.Putting it all together, if p,q are distinct,&amp;nbsp;hh&amp;nbsp;″+(h′)2=0&amp;nbsp;has four distinct roots, two of multiplicity two at p,q and two distinct ones between p and q. If&amp;nbsp;p=q, then&amp;nbsp;hh&amp;nbsp;″+(h′)2&amp;nbsp;has only one root, which however has multiplicity 6.

f(x), g(x), 2 quadratic polynomials: \(\displaystyle{\left|{f{{\left({x}\right)}}}\right|}≥{\left|{g{{\left({x}\right)}}}\right|}∀{x}∈{R}\). Find the

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