Question

(x^3+2x^2-4x-8)/(x^4-16)x(3x^2+8x+5)/3x^2+11x+10)

Answer 1

factor each part
here is the hardest part

x^3 + 2x^2 - 4x - 8 , using grouping
= x^2(x+2) - 4(x+2)
= (x^2 - 4)(x+2)
= (x+2)(x-2)(x+2) or (x-2)(x+2)^2

so...
(x^3+2x^2-4x-8)/(x^4-16)x(3x^2+8x+5)/3x^2+11x+10)
= (x-2)(x+2)^2 /((x-2)(x+2)(x^2 + 4)) * (x+1)(3x+5)/((3x+5)(x+2))
= <b>(x+1)/(x^2 + 4)</b> , x ≠ ±2, -5/3

check by picking any value of x other than the restricted values

e.g. let x = 1
in original:
= (-9)/(-15)*(16/24) = 2/5
value of (x+1)/(x^2+4) = 2/5