How do we middle term split in maths

the thing is that my teacher is taking test today evening so wanna know how to middle term split.
i heard there is a formula to do so

take for example ----------
x^2 - 3x +2
now wehave to split -3 in such a way that-------------
rule ------the two numbers should be such that--
(a) sum of two numbers = -- 3 [ as ( -1 ),+( -- 2) = - 3 ]
(b) product should be = + 2 { -1 x -2 = +2]
SO RULE IS THAT-------------- IF ax^2 +bx + c OR x^2 +b/ax + c/a
the coefficient of x^2 should be unity
for this let two numbers be A &B SUCH THAT - b/a = A+B & c/a = AXB--------------(I)
ANOTHER EXAMPLE-------------- x^2 - 2x - 15----------------------(II)
compare (I) & (II) --2 = --5 +3 so b/a= -2 & A=-5& b= 3 so -- 15 = -5 x3 so c/a = -15
SO THE GENERAL RULE IS THAT ----------
IF A & B are two numbers such that
coefficient of x= A+B & constant quantity = AXB { coefficient of x^2 =unity ]
take another example --------- 2x^2 + 6x + 4
now there are two options for this question ------------
(I) now we have split 6 in such a way that
sum = 6 = { 2 + $}& product = 2x4 =8 [ we have not made the coefficient of x^2 unity ]
so 2x^2 + 4x +2x + 4= ( x +2) (2 x+2)= 2 {x+2}{ x+ 1}
(II) make coefficient of x unity so 2{ x^2 + 3x + 2}
2 [ x^2 +2x +x +2 } = 2 ( x+2) ( x+1)
ANOTHER EXAMPLE -----------
3x^2 - 5x - 8
now we have to split ( 3x -8) = -24 in such away that
(i) sum = -5= ( -8.+3 ) (II) product = -24 ( -8 x 3)
so two numbers are { - 8 & +3 }
so 3x^2 -8x+ 3x -8
= x( 3x -8) +1 ( 3x -8) = ( 3x-8) ( x+1)
NOTE --- in the ax^2 +bx + c
it is not b to be splited but it is c to be splited in such a way ----
that sum of splitted numbers= b & product of two numbers = c

x^2

consider the equation X^2+aX+b=0