What is the number of real roots of the polynomial? What is the number of real roots of the polynomial  $3s^{3}+10s^{2}+14s+8$?

Let $f(s) = 3s^3 + 10s^2 +14s + 8$.  Since the degree of $f$ is $3$ (being odd), you know that there is at least one real root.
Then note that $f'(s) = 9s^2 + 20s + 14$. If you have two real roots, then there would (by the Mean Value Theorem) be some $c$ such that $f'(c) = 0$. But you might be able to show that the equation $f'(s) = 0$ has no real solution.
