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Use synthetic division to determine the quotient and remainder.

(x3 − 41x − 30) ÷ (x + 6)

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Michael J. | Effective High School STEM Tutor & CUNY Math Peer LeaderEffective High School STEM Tutor & CUNY ...
5.0 5.0 (5 lesson ratings) (5)
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The process is the divide the coefficients (descending order of variable) of the dividend by the possible root (divisor).
 
Since -6 is your possible root, we set up the division as this.
 
 
-6 | 1       0         -41         -30
 
              -6          36          30
____________________________
 
      1      -6          -5            0
 
 
As you can see, the digits below the line is the quotient.  The last digit under the line is the remainder. 
 
Quotient = x2 - 6x - 5
Remainder = 0
 
If the reminder is zero, then the quotient is a factor of the dividend.  In this case, we have a remainder of zero.  Therefore,
 
(x2 - 6x - 5) is a factor of (x3 - 41x - 30).
 
Allen E. | Patient and knowledgeble mathematics tutorPatient and knowledgeble mathematics tut...
5.0 5.0 (98 lesson ratings) (98)
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The key value for synthetic division is x + 6 = 0 or x = -6
 
-6|   1     0    -41    -30
              -6    36     30
        1   -6     -5    | 0
 
The bottom line shows the coefficients of the answer with remainder zero
 
Thus (x3 - 41x - 30)/(x + 6) = x2 - 6x - 5