#(5x-9)/(x^2+5x-14)-(3x+4)/(x^2+3x-10)+(x-5)/(x^2+12x+36)#
Before adding and subtracting, we should factorize all the expressions in denominators.
#x^2+5x-14=x^2+7x-2x-14=x(x+7)-2(x+7)=(x-2)(x+7)#
#x^2+3x-10=x^2+5x-2x-10=x(x+5)-2(x+5)=(x-2)(x+5)#
#x^2+12x+36=(x+6)^2#
Note that there GCD is #(x-2)(x+7)(x+5)(x+6)^2#
We may add and subtract like common fractions but noting them to be algebraic expressions.
Hence #(5x-9)/(x^2+5x-14)-(3x+4)/(x^2+3x-10)+(x-5)/(x^2+12x+36)#
= #(5x-9)/((x-2)(x+7))-(3x+4)/((x-2)(x+5))+(x-5)/(x+6)^2#
= #((5x-9)(x+5)(x+6)^2-(3x+4)(x+7)(x+6)^2+(x-5)(x-2)(x+7)(x+5))/((x-2)(x+7)(x+5)(x+6)^2)#
= #(5x^4+76x^3+327x^2+36x-1620-(3x^4+61x^3+436x^2+1236x+1008)+x^4+5x^3-39x^2-125x+350)/((x-2)(x+7)(x+5)(x+6)^2)#
= #(3x^4+20x^3-148x^2-1325x-2278)/((x-2)(x+7)(x+5)(x+6)^2)#
