To find the zeros of the function \( g(x) = 4x^3 + 13x^2 - 37x - 10 \), we can either use graphical methods or algebraic methods (like synthetic division, factoring, or the Rational Root Theorem).
Given that you referenced the graph of the function, we look for points where \( g(x) = 0 \) (where the graph intersects the x-axis).
The provided response options for the zeros are:
- \( x = {-5, -\frac{1}{4}, 2} \)
- \( x = {-10, -5, -\frac{1}{4}, 2} \)
- \( x = {-10, -5, 2} \)
- \( x = {-5, 2} \)
Without the actual graph, we would assume that the solution can be derived directly from these potential roots.
In many situations, polynomial equations like this one could be analyzed using synthetic division or the Rational Root Theorem. However, if analyzing the graph led to determining where the polynomial intersects the x-axis, you would select the option based on that visual assessment.
From the provided choices, if you have determined the graph indicates that \( g(x) \) crosses the x-axis at -5 and 2, you might select option 4: \( x = {-5, 2} \).
If there are additional values seen to have intersections in the graph, you'd select accordingly.
If solving for roots confirms any of these options, you would likewise select those identified values. Remember that real zeros of a polynomial relate to its factors. If you would like, we can perform algebraic factorizations or numerical methods to determine the actual roots. Would that be of assistance?