Where does this answer come from? $2x^2+4x+c-1=0$

Question

Note: I could do a simple algebra solution but the question wants me to answer this question using something that relates to parabola function.

That question is "if $2x^2+4x+c-1=0$ what is c?"
the solution in the answer sheet is about you somehow turn that equation into $4^2-4(2)(c-1) = 0$ which I kinda figure out that it's a rearranged version of $x^2=4fy \to x^2-4fy=0$ but I can't figure out why $x=4$ or why $f=2$ ?

Thanks!

Edit: look like the main problem is that I thought $4^2-4(2)(c-1)=0$ is a rearranged form of $x^2=4fy$ but it's actually a Quadratic's Discriminant $D=b^2-4ac$ which will answer me perfectly where 4 and 2 come from.

Answer 1

$$c=-2x^2-4x+1=-2(x+1)^2+3\leq3.$$ You can use also your idea: $$2^2-2(c-1)\geq0,$$ which gives $c\leq3$ again.