-2x²+5x+12

Since the x² term is negative, we will 
factor out -1:

-1(2x² - 5x - 12)

or just:


-(2x² - 5x - 12)

There is only one way to factor the 2x², as x and 2x
So we write

-(x  )(2x  )

The ways to factor 12 are 12·1, 1·12, 6·2, 2·6, 4·3, 3·4

-(x 12)(2x 1)
-(x 1)(2x 12)
-(x 6)(2x 2)
-(x 2)(2x 6)
-(x 4)(2x 3)
-(x 3)(2x  4)

There was no common factor to take out of the
 3 terms in the parentheses, so we can
eliminate any of those that have a common factor.
So we have only these

-(x 12)(2x 1)
-(x 4)(2x 3)

The only promising one is the second one

-(x 4)(2x 3)

To make the middle term come out as -5x,
we have to put the signs in this way:

-(x-4)(2x+3)

Edwin