  The Eisenbud-Green-Harris (EGH) conjecture states that a homogeneous ideal in
a polynomial ring $K[x_1,\,\ldots,\,x_n]$ over a field $K$ that contains a
regular sequence $f_1,\,\ldots,\, f_n$ with degrees $a_i$, $i=1,\,\ldots,\,n$
has the same Hilbert function as a lex-plus-powers ideal containing the powers
$x_i^{a_i}$, $i=1,\,\ldots,\,n$. In this paper, we discuss a case of the EGH
conjecture for homogeneous ideals generated by $n+2$ quadrics containing a
regular sequence $f_1,\, \ldots, \, f_n$ and give a complete proof for EGH when
$n=5$ and $a_1=\cdots=a_5=2$.
