Profit-maximization for a monopoly
A monopolist maximizes profit. For me, it is usually easier to do this in the quantity space. So you rearange the demand and maximize
$$\max_{Q_p,Q_r} \quad P_p(Q_p)Q_p + P_r(Q_r)Q_r - TC(Q_p+Q_r)$$
$$\max_{Q_p,Q_r} \quad (Q_p-10)Q_p + (28-2Q_r)Q_r - 5-2(Q_r+Q_p) -\frac{(Q_r+Q_p)^2}{8}$$
The FOC gives you two equations with two unknowns, $Q_r$ and $Q_p$, which you solve for. Both FOC equations can be written in the form $MR=MC$. You then plug these $Q$s into the demand equations to get the prices. Next, you calculate consumer surplus and producer surplus and add them to get social welfare.