The correct answer is:  [C]:  "  f(x)  ≤  x²  − x − 1 " .
Explanation:
Note that the graph has a solid line.  This means that the correct equation for the graph contain either:  1)  ≤  or:  2)  ≥ .
We note that all of the answer choices provided have either:  ≤  or:  ≥  .
Let us start with a point on the graph that is "not shaded" (i.e. that is NOT a solution: Â " (0, 2) " ; that is: Â when "x = 0" , y = 2" .
Try this for all the answer choices:
Choice:  [A]:  " y  ≥  x² − x  −  1 " ;
→  " 2  ≥  0² − 0  −  1  " ? ; Â
→  " 2  ≥  0 − 1 " ? ;Â
→  "2  ≥ -1 " ?  Yes!Â
So we can rule out "Choice: [A]" .Â
Try Choice: [B]:
→  " y ≤ −x² − x −1 " ;Â
→  " 2 ≤ −(0²) − 0 − 1 " ?Â
→  " 2 ≤  0 − 0 − 1 " ?
→  " 2 ≤  -1 " ?  No!  So this is a possible solution.
Now, let's try Choice: Â [C]"
→  " y ≤  x²  − x − 1 " ;Â
→  " 2 ≤  0² − 0 − 1 " ? ;
→  " 2 ≤  0 − 1 " ? ;Â
→  " 2 ≤  -1 " ? ;  No!  So this is possible answer.
Now, let's try the last answer choice: Â [D]:
→  " y ≥  −x²  − x − 1 " ;
→  " 2 ≥  −0² − 0 − 1 " ? ;Â
→  " 2 ≥  0 − 0 − 1 " ? ;Â
→  " 2 ≥  - 1 " ? ;  No!  So this is a possible answer!
Now, we have narrow our choices to B, C, and D.
Now , among these choices, which of these fits a point with the shaded region?
Let us pick a point: Â "(2, 0) " Â that, is, when "x = 2, y = 0 " .
Let us start with: Â "Choice: Â [B] " :
→  " y ≤ −x² − x −1 " ;Â
→  " 0  ≤ −(2²) − 2 − 1 " ?Â
→  " 0 ≤  -4 − 2 − 1 " ?
→  " 0 ≤  -7 " ?  No!  So we can rule out "Choice B" .Â
Now, let us try "Choice: Â [C]" :
→  " y ≤  x²  − x − 1 " ;Â
→  " 0 ≤  2² − 2 − 1 " ? ;
→  " 0 ≤  4 − 2  − 1 " ? ;Â
→  " 0  ≤  1 " ? ;  Yes!  So; this should be the correct
Answer choice:  [C]:  " f(x)  ≤  x²  − x − 1 " ;Â
Let us try " Choice:  [D]"— to confirm that "Choice: [D]" is incorrect:
→  " 0  ≥  −(2²)  − 2 − 1 " ;
→  " 0  ≥  −(2²) −  2 − 1 " ? ;Â
→  " 0  ≥  − 4  −  2 − 1 " ? ;Â
→  " 0  ≥  − 7 " ? ;  No!  So we can rule out: "Choice:  D" .
The correct answer is:  [C]:  "  f(x)  ≤  x²  − x − 1 " .