  The aim of this paper consists on the study of the following fourth-order
operator: \begin{equation}\label{Ec::T4} T[M]\,u(t)\equiv
u^{(4)}(t)+p_1(t)\,u"'(t)+p_2(t)\,u"(t)+M\,u(t)\,,\ t\in I \equiv [a,b]\,,
\end{equation} coupled with the two point boundary conditions:
\begin{equation}\label{Ec::cf} u(a)=u(b)=u"(a)=u"(b)=0\,. \end{equation}
  So, we define the following space:
  \begin{equation}\label{Ec::esp}
  X=\left\lbrace u\in C^4(I)\quad\mid\quad u(a)=u(b)=u"(a)=u"(b)=0
\right\rbrace \,.
  \end{equation}
  Here $p_1\in C^3(I)$ and $p_2\in C^2(I)$.
  By assuming that the second order linear differential equation
  \begin{equation}\label{Ec::2or}
  L_2\, u(t)\equiv u"(t)+p_1(t)\,u'(t)+p_2(t)\,u(t)=0\,,\quad t\in I,
  \end{equation} is disconjugate on $I$, we characterize the parameter's set
where the Green's function related to operator $T[M]$ in $X$ is of constant
sign on $I \times I$. Such characterization is equivalent to the strongly
inverse positive (negative) character of operator $T[M]$ on $X$ and comes from
the first eigenvalues of operator $T[0]$ on suitable spaces.
