Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 5 / a / Solution

Writing the polynomial as $p(x)$, direct evaluation gives $p(0)=-6$ and $p(3)=42$. A polynomial is continuous, so the intermediate value theorem guarantees a polynomial root between them. Trying the integer $2$ gives $16+12-22-6=0$, hence $x=2$ is a root.