Explanation:
so, if I understand your text correctly, we need to bring
2 - (x+2)/(x-3) - (x-6)/(x+3)
into a form
(ax + b)/(x² - 9)
what we notice immediately is
(x+3)(x-3) = x² - 9
that makes sense, as we want to transform all terms into fractions with the same denominator.
and the necessary "criss-cross" multiplication leads to (x² -9) as denominator.
so, let's transform every term to a fraction with that denominator, and then we add or subtract them all up as per the original expression.
2 :
multiply by (x²-9)/(x²-9)
2(x²-9)/(x²-9) = (2x²-18)/(x²-9)
(x+2)/(x-3) :
multiply by (x+3)/(x+3)
(x+2)(x+3)/(x²-9) = (x²+3x+2x+6)/(x²-9) =
= (x²+5x+6)/(x²-9)
(x-6)/(x+3) :
multiply by (x-3)/(x-3)
(x-6)(x-3)/(x²-9) = (x²-3x-6x+18)/(x²-9) =
= (x²-9x+18)/(x²-9)
for the whole expression we get then
(2x²-18)/(x²-9) - (x²+5x+6)/(x²-9) - (x²-9x+18)/(x²-9) =
= (2x²-x²-x² -5x+9x -18-6-18)/(x²-9) =
= ( 0 + 4x - 42 )/(x²-9)
a = 4
b = -42