The problem asks us to divide the polynomial $f(\alpha) = \alpha^4 + 5\alpha^3 - 3\alpha^2$ by $g(\alpha) = 5\alpha^2 - \alpha + 2$ using the division algorithm to find the quotient $q(\alpha)$ and remainder $r(\alpha)$.

Let's apply the polynomial division process.

We will divide $f(\alpha)$ by $g(\alpha)$ and express $f(\alpha)$ as: $f(\alpha) = g(\alpha) \cdot q(\alpha) + r(\alpha)$ where the degree of the remainder $r(\alpha)$ will be less than the degree of $g(\alpha)$.

I'll now proceed with the division process to compute the quotient and remainder.