Confused on factoring when 'a' is greater than 1

[Chaim]

For example:
f(x)=2x²+5x-3
So I did the tic tac toe method which the teacher taught us
ax² c ax²c


Which then I plug it in
2x² -3 -6x²
2x -1 -2x
1x 3 3x

Then circle diagonally at the 2x2 square at the bottom left
Which I get (2x+3)(1x-1)
Which was apparently wrong

It's suppose to be
2x² -3 -6x²
2x 3 6x
1x -1 -1x
Therefore I get (2x-1)(1x+3)
My zeroes would then be (1/2) and -3

I'm kind of confuse what made the first one wrong
And is there another way to factor? Cause you would have to find a lot of multiples that would fit.

 

[Chaim]

Are you familiar with the quadratic formula?

Oh right!
At first I didn't get it because it meant finding the values of x
But I could actually use that to factor it out
Thanks