Partial Fraction Decomposition

for decomposition to fractions, firstly we need to find roots of equation 6x ²+7x-3=0
Discriminant of equation
D=49+72=121=11 ²
roots:
x _1/2=(-7±11)/12= -3/2 and 1/3
so, second-degree polynomial 6x ²+7x-3 can be written as 6(x+3/2)(x-1/3) or (2x+3)(3x-1)
from there fraction 22/(6x^2+7x-3)=22/(2x+3)(3x-1) can be written as
22/(2x+3)(3x-1)=a/(2x+3)+b/(3x-1)
now let's find a and b
a/(2x+3)+b/(3x-1)=(a(3x-1)+b(2x+3))/(2x+3)(3x-1)=(3ax-a+2bx+3b)/(2x+3)(3x-1)=(x(3a+2b)+3b-a)/(2x+3)(3x-1)
here we get set of equations:
3a+2b=0
3b-a=22

3a+2b=0
a=3b-22

3(3b-22)+2b=0
a=3b-22

9b-66+2b=0
a=3b-22

11b=66
a=3b-22

b=6
a=3*6-22=-4

so, fraction decomposition of 22/(6x^2+7x-3) will be 6/(3x-1)-4/(2x+3)