The factors of f(x) are (x + 4) , (x - 3) , (x² + 1)
The values of x = -4 and 3
Step-by-step explanation:
* At first we calculate f(-4)
- If it will be = 0
∴ x + 4 is a factor of f(x)
* In the synthetic calculation we use the coefficient of the dividend
 with the factors of the function
∵
∵ x = -4
Step 1 : Write down the coefficients of the f(x) , put x = -4 at the left
             -4     1   1   -11   1   -12
                 ________________
Step 2 : Bring down the first coefficient to the bottom row.
            -4     1   1   -11   1   -12
                 ________________
                  1
Step 3 : Multiply it by -4, and carry the result into the next column.
            -4     1   1   -11   1   -12
                 __-4_____________
                  1
Step 4 : Add down the column
            -4     1   1   -11   1   -12
                 __-4_____________
                  1  -3
Step 5 : Multiply it by -4, and carry the result into the next column
            -4     1   1   -11   1   -12
                 __-4__12___________
                  1  -3
Step 6 : Add down the column
           -4     1   1   -11   1   -12
                 __-4__12___________
                  1  -3   1
Step 7 : Multiply it by -4, and carry the result into the next column
          -4     1   1   -11    1   -12
                __-4__12__-4_________
                1  -3   1
Step 8 : Add down the column
          -4     1   1   -11    1   -12
                __-4__12__-4_________
                1  -3   1    -3
Step 9 : Multiply it by -4 , and carry the result into the next column
         -4     1   1   -11    1   -12
               __-4__12__-4__12_______
               1  -3   1    -3
Step 10 : Add down the column
         -4     1   1   -11    1   -12
               __-4__12__-4__12_______
               1  -3   1    -3   0
∴ The quotient is (x³ - 3x² + x - 3) and the remainder is 0
* At second we calculate f(3)
- If it will be = 0
∴ x - 3 is a factor of f(x)
* We will do the same steps with x = 3
∵ f(x) = x³ - 3x² + x - 3
∵ x = 3
Step 1 : Write down the coefficients of the f(x) , put x = 3 at the left
             3     1   -3   1   -3 Â
                 ____________
Step 2 : Bring down the first coefficient to the bottom row.
            3     1   -3   1   -3  Â
                _____________
                 1
Step 3 : Multiply it by 3, and carry the result into the next column.
            3     1   -3   1   -3  Â
                ____3________
                 1
Step 4 : Add down the column
            3     1   -3   1   -3 Â
                 ___3____________
                 1   0
Step 5 : Multiply it by 3, and carry the result into the next column
            3     1   -3   1   -3 Â
                 ___3__0___
                  1   0
Step 6 : Add down the column
            3     1   -3   1   -3 Â
                 ___3__0___
                 1   0   1
Step 7 : Multiply it by 3, and carry the result into the next column
           3     1   -3   1   -3 Â
                ___3__0__3_
                 1   0   1  Â
Step 8 : Add down the column
           3     1   -3   1   -3 Â
                ___3__0__3_
                 1   0   1   0
∴ The quotient is (x² + 1) and the remainder is 0
* The factors of f(x) are (x + 4) , (x - 3) , (x² + 1)
∴ The values of x = -4 and 3