With the polynomial $x^4 -7x^3 + 3x^2 + 31x + 20$ where $(x+1)$ is a factor for example.
Do I use the same procedure but start a degree higher, like $(x+1)(\alpha x^3 + \beta x^2 + \gamma x + \delta)$ and move on down to figure out the 2nd degree after that?
There are other techniques, namely the polynomial long division as mentioned in a comment by pedja, the Ruffini's rule and for polynomials with integer coefficients the rational root theorem, but you can apply the method of equating coefficients to
$\begin{eqnarray*} P(x) &:&=x^{4}-7x^{3}+3x^{2}+31x+20 =\left( x+1\right) \left( \alpha x^{3}+\beta x^{2}+\gamma x+\delta \right). \end{eqnarray*}\;\; \tag{1}$
Expanding the RHS and comparing with the LHS, where the coefficient of $x^{4}$ is $1$, you would conclude that $\alpha =1$ and are left with a simpler equality
$\begin{eqnarray*} P(x) &:&=x^{4}-7x^{3}+3x^{2}+31x+20=\left( x+1\right) \left( x^{3}+\beta x^{2}+\gamma x+\delta \right) \\ &=&x^{4}+\left( \beta +1\right) x^{3}+\left( \beta +\gamma \right) x^{2}+\left( \gamma +\delta \right) x+\delta . \end{eqnarray*}$ Equating coefficients you get the simple system of four linear equations $ \begin{equation*} \left\{ \begin{array}{c} \delta =20 \\ \gamma +\delta =31 \\ \beta +\gamma =3 \\ \beta +1=-7, \end{array} \right. \end{equation*}$ whose solution is $\delta =20,\beta =-8,\gamma =11$. So
$\begin{equation*} x^{4}-7x^{3}+3x^{2}+31x+20=\left( x+1\right) \left( x^{3}-8x^{2}+11x+20\right). \end{equation*}\tag{2}$ Now by direct inspection (or by the rational root theorem, since $1,2,4$ and $5$ are the positive divisors of $20$) you can find that $x=-1$ is a zero of the cubic polynomial $Q(x):=x^{3}-8x^{2}+11x+20$, ie $Q(-1)=0$. Applying a similar method to $Q(x)$ you would find that
$\begin{equation*} Q(x)=x^{3}-8x^{2}+11x+20=\left( x+1\right) \left( x-4\right) \left( x-5\right) . \end{equation*}$
Consequently, $\begin{equation*} x^{4}-7x^{3}+3x^{2}+31x+20=\left( x+1\right) ^{2}\left( x-4\right) \left( x-5\right). \end{equation*}\tag{3}$
A shorter method is the Ruffini's rule applied to the polynomial division of $P(x)$ by $(x-r)$. This case (with $P(x)=x^{4}-7x^{3}+3x^{2}+31x+20$ and $r=-1,P(r)=0$, ) is shown bellow, where the remainder is $P(r)=s$.
