Answer:
±2
Explanation:
For simplicity let's take a look at a general third order polynomial:
ax^3 + bx^2 + cx + d. In this particular case,
rational roots have the form ±d/±a. Note that these are fractions/ratios. Beyond that, we factor d and choose numerators ±(all possible whole number factors of d, dividing these results by all possible factors of a.
Looking at 4x^3 + 9x^2 - x + 10 = 0, we see that d = 10 and that factors of d include ±1, ±2, ±5 and ±10. a = 4 and factors of a include ±1, ±2, ±4.
So, any rational roots of the given polynomial will stem from the possible rational roots
±1 ±2 ±5 ±10 ±1 ±2 ±5 ±10
--- , ----- , -----, ------- , ---- , ----- , --- , ------ , and so on,
±1 ±1 ±1 ±1 ±2 ±2 ±2 ±2
until you have used up all of the possible factors of 10 and all of the possible factors of 4.
Of the four possible rational roots you have shared, only ±2 (which would actually be ±2 / ±1) is acceptable.