can it be solved without vieta formulae ??

Question

recently i came across an anonymously remarkable algebra question which is as follows If the polynomial $$F(x)= 4x^4 - ax^3 + bx^2 - cx + 5$$ where $a,b,c$ belongs to $\mathbb R$ has 4 positive real zeroes(roots) say $M,N,P,L$ such that $$M/2 + N/4 + P/5 + L/8 = 1$$ then find the value of $a$, by using AM-GM inequality & the Vieta formulas i got $a=19$; what i am curious is if this could be solved without using vieta formulas i tried a lot but with no fruits, So can anyone tell ?