  We prove two conjectures of Br\"{a}nd\'{e}n on the real-rootedness of
polynomials $Q_n(x)$ and $R_n(x)$ which are related to the Boros-Moll
polynomials $P_n(x)$. In fact, we show that both $Q_n(x)$ and $R_n(x)$ form
Sturm sequences. The first conjecture implies the 2-log-concavity of $P_n(x)$,
and the second conjecture implies the 3-log-concavity of $P_n(x)$.

Combinatorics;Classical Analysis and ODEs