$t$ and $s(t)$ are physical quantitities. Quoted from [Wikipedia][1]:

> A physical quantity can be expressed as a *value*,
  which is the algebraic multiplication of a 
  *numerical value* and a *unit of measurement*. 

Your equation 
$$s(t) = 2t^3 - 14t^2 + 20t - 44$$
is not correct to begin with.
$s(t)$ is a length (having unit $\text{m}$)
and $t$ is a time (having unit $\text{s}$).
Hence the equation above is [dimensionally inhomogeneous][2].

You make it dimensionally homogeneous by writing
$$s(t) = 2\frac{\text{m}}{\text{s}^3}\ t^3
 - 14\frac{\text{m}}{\text{s}^2}\ t^2
 + 20\frac{\text{m}}{\text{s}}\ t - 44\text{ m}$$

When you now insert a value for $t$ with unit $\text{s}$
(for example $t=4.2\text{ s}$),
you see the $\text{s}^3$, $\text{s}^2$ and $\text{s}$
in the terms on the right side all cancel out,
and you finally get a number with unit $\text{m}$
which matches the unit of $s(t)$ on the left side.


  [1]: https://en.wikipedia.org/wiki/Physical_quantity
  [2]: https://en.wikipedia.org/wiki/Dimensional_analysis#Dimensional_homogeneity_(commensurability)