Let $\{P_n \}_{n\ge0}$ be a sequence of monic orthogonal polynomials with
respect to a quasi--definite linear functional $u$ and $\{Q_n \}_{n\ge0}$ a
sequence of polynomials defined by $$Q_n(x)=P_n(x)+s_n P_{n-1}(x)+t_n
P_{n-2}(x),\quad n\ge1,$$ with $t_n \not= 0$ for $n\ge2$.

We obtain a new characterization of the orthogonality of the sequence $\{Q_n
\}_{n\ge0}$ with respect to a linear functional $v$, in terms of the
coefficients of a quadratic polynomial $h$ such that $h(x)v= u$.

We also study some cases in which the parameters $s_n$ and $t_n$ can be
computed more easily, and give several examples.

Finally, the interpretation of such a perturbation in terms of the Jacobi
matrices associated with $\{P_n \}_{n\ge0}$ and $\{Q_n \}_{n\ge0}$ is
presented.