To find the zeros of the polynomial function \( g(x) = 4x^3 + 13x^2 - 37x - 10 \), we are looking for the values of \( x \) where \( g(x) = 0 \).
Based on the description of the graph, which has turning points and crosses the x-axis, we can infer the approximate locations of the zeros. However, we can also directly check the potential roots given in the options provided.
One efficient way to find the zeros is to use synthetic division, the Rational Root Theorem, or graphing the function to identify where it crosses the x-axis.
The options for the zeros provided are:
- \( x = {-10, -5, -14, 2} \)
- \( x = {-5, 2} \)
- \( x = {-5, -\frac{1}{4}, 2} \)
- \( x = {-10, -5, 2} \)
Based on the general shape of a cubic function and the provided graph information:
- Since the function rises in quadrant 3, turns in quadrant 2, and drops to another turning point in quadrant 4 before rising again through quadrant 1, it's reasonable to expect some zeros to be present in the positive and negative x-values.
From inspection and potential evaluation:
- \( g(-5) = 4(-5)^3 + 13(-5)^2 - 37(-5) - 10 = 0 \) (this is a zero)
- \( g(2) = 4(2)^3 + 13(2)^2 - 37(2) - 10 = 0 \) (this is a zero)
For the options provided, the zeros would likely be:
- From the first option, \( -10 \) and \( -14 \) seem unjustifiable based on the polynomial's graph as described.
After examining the valid choices, the most reasonable pair of answers based on derived values after calculations and the graph's intersection is:
x = {-5, 2}
So, the correct response would be: x = {-5, 2}