Question

What is the difference?

StartFraction 2 x + 5 Over x squared minus 3 x EndFraction minus StartFraction 3 x + 5 Over x cubed minus 9 x EndFraction minus StartFraction x + 1 Over x squared minus 9 EndFraction
StartFraction (x + 5) (x + 2) Over x cubed minus 9 x EndFraction
StartFraction (x + 5) (x + 4) Over x cubed minus 9 x EndFraction
StartFraction negative 2 x + 11 Over x cubed minus 12 x minus 9 EndFraction
StartFraction 3 (x + 2) Over x squared minus 3 x EndFraction

Answer 1

Compute with common denominator x(x-3)(x+3) = x^3-9x.

(2x+5)/(x^2-3x) = (2x+5)(x+3)/[x(x-3)(x+3)]
(3x+5)/(x^3-9x) = (3x+5)/[x(x-3)(x+3)]
(x+1)/(x^2-9) = x(x+1)/[x(x-3)(x+3)]

So the numerator is
(2x+5)(x+3) - (3x+5) - x(x+1)
= (2x^2+11x+15) - (3x+5) - (x^2+x)
= x^2+7x+10
= (x+5)(x+2).

Thus the difference simplifies to (x+5)(x+2)/(x^3-9x). (Domain: x ≠ 0, ±3.)

This matches the first choice.