$p(x)$ will be a multiple of $x+2,$ only if $x+2$ divides $p(x)$ leaving remainder zero.

Now, taking $x+2=0,$ we have $x=-2$

Also, $p(-2)=(-2)^{3}+9(-2)^{2}+26(-2)+24$

$=(-8)+9(4)-52+24$

$=-8+36-52+24$

$=-60+60$

$=0$

So, the remainder obtained on dividing $p(x)$ by $x+2$ is $0 .$

So, $(x+2)$ is a factor of the given polynomial $p(x),$ that is $p(x)$ is a multiple of $x+2$