Show that p (x) = 2x^3 - 5x^2 - 10x + 5 has a real root.

All odd degrees polynomials with real coefficients have (at least) a real root, and are continuous. This is because the curve goes diagonally and must pass through the x-axis.

The above polynomial can be evaluated at x1=-10 and x1=+10 (or any other large enough number)

f (-10) = - 2395

f (10) = 1405

Since they have opposite signs, the function must intersect the x-axis between x1 and x2 by the intermediate value theorem, hence there is (at least) one root.