Answer
$$\color {blue}{\bf\text{Yes, the points make a right triangle}}$$
Work Step by Step
For clarity lets start by naming our points:
$A=(-7,4)$
$B=(6,-2)$
$C=(0,-15)$
First, we'll use the distance formula to find the lengths of each side:
$d(P,Q)=\sqrt{(x_{P}-x_{Q})^2+(y_{P}-y_{Q})^2}$
$AB= \sqrt{(-7-6)^2+(4-(-2))^2}$
$AB= \sqrt{(-13)^2+6^2}$
$AB= \sqrt{169+36}$
$AB= \sqrt{205}$
$AC= \sqrt{(-7-0)^2+(4-(-15))^2}$
$AC= \sqrt{(-7)^2+19^2}$
$AC= \sqrt{49+361}$
$AC= \sqrt{410}$
$BC= \sqrt{(6-0)^2+(-2-(-15))^2}$
$BC= \sqrt{6^2+13^2}$
$BC= \sqrt{36+169}$
$BC= \sqrt{205}$
Now that we have the lengths of the sides,
$AB= \sqrt{205}$, $AC= \sqrt{410}$, $BC= \sqrt{205}$,
we can apply the Pythagorean Theorem $a^2+b^2=c^2$ to see if the sides make a right triangle.
$(\sqrt{205})^2+(\sqrt{205})^2=(\sqrt{410})^2$
$205+205=410$
$410=410$
Which is $\bf \text{true}$, so:
$$\color {blue}{\bf\text{Yes, the points make a right triangle}}$$
