Which of the following graphs represents the function f(x) = −2x3 − x2 + 3x + 1?

graph with 3 real zeros, down on left, up on right
graph with 3 real zeros, up on left, down on right
graph with 2 real zeros, down on left, down on right
graph with 2 real zeros, up on left, up on right

The given function is f(x) = -2x³ - x² + 3x + 1.

Obtain f(-x) as
f(-x) = 2x³ - x² - 3x + 1.

Observations:
1. This a 3rd order polynomial. It can have at most 3 real zeros.
2. The coefficient of the leading term of the polynomial is negative, therefore
the curve goes up on the left,  and down on the right.

According to Descartes' Rule of Signs:
3. f(x) has one change in sign, so there is one positive real root.
4. f(-x) has two changes in sign, so there are 2 negative real roots.

Graph with 3 real roots, up on left, down on right.