Answer
$$\color {blue}{\bf\text{Yes, the points are collinear}}$$
Work Step by Step
For clarity lets start by naming our points:
$A=(0,-7)$
$B=(-3,5)$
$C=(2,-15)$
First, we'll use the distance formula for the distance between each pair of points:
$d(P,Q)=\sqrt{(x_{P}-x_{Q})^2+(y_{P}-y_{Q})^2}$
$AB= \sqrt{(0-(-3))^2+(-7-5)^2}$
$AB= \sqrt{3^2+(-12)^2}$
$AB= \sqrt{9+144}$
$AB= \sqrt{153}$
$AB= 3\sqrt{17}$
$AC= \sqrt{(0-2)^2+(-7-(-15))^2}$
$AC= \sqrt{(-2)^2+8^2}$
$AC= \sqrt{4+64}$
$AC= \sqrt{68}$
$AC= 2\sqrt{17}$
$BC= \sqrt{(-3-2)^2+(5-(-15))^2}$
$BC= \sqrt{(-5)^2+20^2}$
$BC= \sqrt{25+400}$
$BC= \sqrt{425}$
$BC= 5\sqrt{17}$
Now that we have the lengths of the sides,
$AB= 3\sqrt{17}$, $AC= 2\sqrt{17}$, $BC=5\sqrt{17}$,
we can see if the two shortest equal the longest.
$3\sqrt{17} +2\sqrt{17} = 5\sqrt{17}$
Which is $\bf \text{true}$, so:
$$\color {blue}{\bf\text{Yes, the points are collinear}}$$
