Reducing "multiple satisfiability" to normal SAT
CNF-SAT can be reduced to QUADRUPLE-SAT by adding two new variables, $x_1$ and $x_2$, and adding the following CNF clauses.

$(x_1 \lor \lnot x_1)$

$(x_2 \lor \lnot x_2)$

Because $x_1$ and $x_2$ can take any value and not affect satisfiability, if the original formula had at least one solution then the new formula will have at least four.  And if the original formula had no solutions then the new formula will still have none.

Since CNF-SAT is NP-complete, then this reduction to QUADRUPLE-SAT proves the NP-completeness of the QUADRUPLE-SAT.