Let P(x) be a polynomial with degree 3, consider the polynomial

Q(x)=(x^{3}-2x+1-P(x))(2x^{3}-5x^{2}+4-P(x)).

Assume that Q(x)≤0, ∀x and P(0)=3. Calculate Q(-1).

(In reply to

re(2): Solution by Larry)

I was suspicious because the difference was a cubic, and a cubic has 1 or 3 real roots. Of course, if it has 3 real roots, then 2 or 3 of them can be identical, so I had to check to see if the 3 real roots were distinct, which they were.