If $PQ=PR$, then the angles at $Q$ and $R$ (the base angles) are equal.

After you know the angles at $Q$ and $R$, subtract them from $180^\circ$ to get the angle at $P$.

In Desmos, define

q = 180 - 146

Because the base angles are both $q$, enter

p = 180 - 2q

Desmos will evaluate $p$, which is the measure of $\angle QPR$. The correct choice matches this value.

Because $\overline{PQ}$ is extended through $Q$ to $S$, $\angle SQR$ and $\angle PQR$ form a linear pair (they are supplementary).

So,

Given $PQ = PR$, $\triangle PQR$ is isosceles with base $\overline{QR}$. Therefore, the base angles are equal:

The angles of a triangle sum to $180^\circ$:

So the correct choice is $112^\circ$.