Example 6 (Wang). Find the locus of the vertex of the parabola $y=x^{2}-4 t^{2} x$ $+\left(4 t^{4}-t^{2}-1\right)$ (t is a parameter).
Let the point be $(\mathrm{x}, \mathrm{y})$, then
$$
\left\{\begin{array}{l}
x=2 t^{2} \\
y=-\left(t^{2}+1\right)
\end{array}\right.
$$

Eliminating the parameter, we get $x+2 y+2=0$, which is the line.

Analysis: Due to the neglect of $t^{2} \geqslant 0$, i.e., $x \geqslant 0$, the solution is incomplete. In fact, the trajectory is the ray $x+2 y+2=0$ $(x \geq 0)$.