#### Answer

The curve intersects the x-axis when $ t=3$.

#### Work Step by Step

The space curve given by
$$ r(t) = \langle t^2, t^2-2t-3, t-3 \rangle $$
intersects the x-axis when $ y=0, z=0$; that is, we have
$$ t^2-2t-3=0, \quad t-3=0\Longrightarrow t=3.$$
Hence, the curve intersects the x-axis when $ t=3$.