To find out the number of components of a given set

Question

Find the number of connected components of the set $$\left\{x\in \mathbb R : x^3\left(x^2+5x-\frac{65}{3}\right)>70x^2-300x-297\right\}$$ under the usual topology on $\mathbb R$.

I tried to factorize the both right hand side and left hand side but fails. It is difficult to handle the too large constant. What's the actual process to solve this type of problems ?

Any hint. ?

Answer 1

I would find the real roots of $f(x)=x^3(x^2+5x-65/3)-70x^2+300x+297$ first which would divide $\mathbb{R}$ into a maximum of $6$ subintervals. Then check individually inside each subinterval for $f(x)>0$. Those subintervals would be the connected components of this set.