Answer
$$-16$$
Work Step by Step
We know that the line equation can be defined as: $$ r_0+kt=(0,0,0)+t \space \lt 1,2,3 \gt=\lt t, 2t, -3t \gt \\
x=t \implies dx= dt \\ y=2 \space t \implies dy=2 \space dt \\z=3t \implies dz=3 \space dt $$
Substitute all the above values in the given integral as follows:
$$
\int_{0,0,0}^{1,2,3} \space (4t^2)dt+2(t-9t^2) \space dt-(2)(2t)(3t) \space 3 \space dt=-16$$
