How do I use the intermediate value theorem to determine whether x5+3x2−1=0 has a solution...

Explanation: 
According to the intermediate value theorem:
If the function ${\displaystyle f\left(x\right)}$ is continuous on the interval ${\displaystyle [a,b]}$ and ${\displaystyle y}$ is a number between ${\displaystyle f\left(a\right)\text{ and }f\left(b\right)}$, then there exists a number ${\displaystyle x=c}$ in the open interval ${\displaystyle (a,b)}$ such that ${\displaystyle y=f\left(c\right)}$. 
Here, 
${\displaystyle f\left(x\right)=x^5+3x^2-1}$ 
is a continouos function on the interval ${\displaystyle [0,3]}$ as ${\displaystyle f\left(x\right)}$ is a polynomial function. 
${\displaystyle f\left(0\right)=-1}$ 
${\displaystyle f\left(3\right)=243+27-1=269}$ 
${\displaystyle 0\in (f\left(0\right),f\left(3\right))}$ 
As ${\displaystyle f\left(x\right)}$ changes sign from ${\displaystyle f\left(0\right)}$ as ${\displaystyle f\left(3\right)}$ 
${\displaystyle \mathrm{\exists }c\in (0,3)}$ such that ${\displaystyle f\left(c\right)=0}$ 
The intermediate value theorem is used in this situation.