Question

how do i prove there must be a root between (-2,0) for X^4+5x^3+6?

Answer 1

\[f(x)=x^4+5x^3+6\]\[f(0)=6\]\[f(-2)=16+5\cdot (-8)+6=-18\]\[f(-2)\cdot f(0)<0\quad\Rightarrow\quad\mbox{at least one root beetwen -2 and 0.}\]

Answer 2

Use the intermediate value theorem