Question

x = t2 + 3, y = t2 - 2
eliminate the parameter

Answer 1

is just a matter of a system of equations and substitution

$\bf \begin{cases} x=t^2+3\implies x-3=\boxed{t^2}\\ y=t^2-2\\ ------------\\ y=\boxed{x-3}-2\\ y=x-5 \end{cases}$

Answer 2

So, you have to get rid of t^2, get it in the first one:

t^2 = x-3,

Now you can go to the other equation: y = t^2 - 2 = (x-3)-2=x-5

It's a line indeed!