The question is
Let $n$ be a positive integer. Show that there are positive real numbers $a_0,a_1,\dots, a_n$ such that for each choice of signs the polynomial $$\pm a_{n} x^{n} \pm a_{n-1} x^{n-1}\pm a_{n-2} x^{n-2 }+\cdots \pm a_{1} x^{1}\pm a_{0} x^{0}$$ has $n$ distinct real roots.
I cannot determine a condition to guarantee $n$ distinct real roots for a polynomial.
One way for getting multiple roots is to use intermediate value theorem to produce at least one root in $(0,1),(1,2),(2,3), \dots, (n-1,n)$, but we are changing signs of coefficient for the rest of polynomials so I am not able to proceed from here.