Are Roman Catholics required to believe that the Cardinals choice of a new Pope is inspired by the Holy Spirit?
If so, how do we explain "bad" popes?
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The Holy Spirit should be invoked and should guide the election of the Pope.
From Bl. John Paul II's Apostolic Constitution:
[Cardinals over the age of eighty] in particular should lead the People of God assembled in the Patriarchal Basilicas of Rome and in other churches in the Dioceses throughout the world, supporting the work of the electors with fervent prayers and supplications to the Holy Spirit and imploring for them the light needed to make their choice before God alone and with concern only for the "salvation of souls, which in the Church must always be the supreme law"
In view of the sacredness of the act of election and thus the need for it to be carried out in an appropriate setting where, on the one hand, liturgical actions can be readily combined with juridical formalities, and where, on the other, the electors can more easily dispose themselves to accept the interior movements of the Holy Spirit, I decree that the election will continue to take place in the Sistine Chapel.
From the Pauline Chapel of the Apostolic Palace, where they will assemble at a suitable hour in the afternoon, the Cardinal electors, in choir dress, and invoking the assistance of the Holy Spirit with the chant of the Veni Creator, will solemnly process to the Sistine Chapel of the Apostolic Palace, where the election will be held.
That said, there remains the possibility that the cardinals will fail to discern the will of the Spirit. However, that can no more be determined by the faithful than it can be by the cardinals themselves. So yes, we must believe that the right person has been chosen.
DJClayworth is correct that, even if the wrong person is elected, his ministry is not invalidated. God will find a way of making everything good.
One of the principles of denominations which give importance to the sacraments is that the unworthiness of the priest does not invalidate the sacrament. Even if your priest has himself sinned, that does not prevent the grace of God acting through that priest in the sacraments. Though I can't find a clear statement about it I would expect a similar principle to apply to popes - that God can use an imperfect pope to perfectly guide his church.