The force acting on a particle at a given time $ t $ is defined as $ F = \alpha t^2 + \beta t $, where $ \alpha $ and $ \beta $ are constants. We need to determine which factor is dimensionless if $ \alpha $ and $ \beta $ are constants.

Using the principle of homogeneity of dimensions:

$ \begin{aligned} & [F] = [\alpha t^2] = [\beta t] \\ & [\alpha] = \frac{[F]}{[t^2]} \quad \text{and} \quad [\beta] = \frac{[F]}{[t]} \\ & \therefore \quad [\alpha][t] = [\beta] \\ & \therefore \quad \frac{\alpha t}{\beta} = \text{dimensionless} \end{aligned} $