#### Answer

$-1, 0, \frac{2}{3}, \frac{5}{6}, 1$

#### Work Step by Step

Subtract 2 to both sides of the inequality to obtain:
$x^2+2-2\lt 4-2
\\x^2 \lt 2$
Thus, the numbers that will satisfy the given inequality are the ones whose squares are less than $2$.
Note that among the elements of $S$, the following have squares that are less than 2:
$(-1)^2=1
\\0^2=0
\\(\frac{2}{3})^2=\frac{4}{9}
\\(\frac{5}{6})^2=\frac{25}{36}
\\1^2=1$
Therefore, the elements of $S$ that satisfy the given inequality are:
$-1, 0, \frac{2}{3}, \frac{5}{6}, 1$