The lines $\frac{x+3}{-4}=\frac{y-6}{3}=\frac{z}{2}$ and $\frac{x+2}{-4}=\frac{y}{1}=\frac{z-7}{1}$ are:

Step 1: Check directions. $\overrightarrow{B_1}=(-4,3,2)$ and $\overrightarrow{B_2}=(-4,1,1)$. They are not proportional, so lines are not parallel.Step 2: Check for intersection using Shortest Distance $d$.Points $A_1=(-3,6,0)$ and $A_2=(-2,0,7)$. $\overrightarrow{A_2}-\overrightarrow{A_1}=(1,-6,7)$.Given $\overrightarrow{B_1}\times \overrightarrow{B_2}=(1,-4,8)$.Magnitude $|\overrightarrow{B_1}\times \overrightarrow{B_2}|=\sqrt{1^2+(-4{)}^2+8^2}=\sqrt{81}=9$.$d=\frac{|(A_2-A_1)\cdot (B_1\times B_2)|}{|\overrightarrow{B_1}\times \overrightarrow{B_2}|}=\frac{|1(1)+(-6)(-4)+7(8)|}{9}=\frac{|1+24+56|}{9}=\frac{81}{9}=9$.Since $d\ne 0$ and they aren't parallel, they are skew lines.