Question

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

Answer 1

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.