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 inconsistent.
You make it dimensionally consistent 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.