Use synthetic division to find the quotient and remainder when

\[ f(x) = 6x^4 - 8x^3 - 5x^2 - 6x + 6 \]

is divided by

\[ g(x) = x - 6 \]

a) Quotient is $6x^3 + 28x^2 + 163x + 978$, Remainder is $-582$

b) Quotient is $6x^3 - 28x^2 + 163x - 978$, Remainder is $582$

c) Quotient is $6x^3 + 28x^2 - 163x - 978$, Remainder is $-582$

d) Quotient is $6x^3 - 28x^2 - 163x + 978$, Remainder is $582$

Using synthetic division, we determined the quotient to be [tex]6x^3 + 28x^2 + 163x + 978[/tex] and the remainder to be -582 when dividing f(x) by g(x). a) Quotient is 6x³ + 28x² + 163x + 978, Remainder is -582

To find the quotient and remainder of[tex]f(x) = 6x^4 - 8x^3 - 5x^2 - 6x + 6[/tex] when divided by g(x) = x - 6 using synthetic division, we follow these steps:

Write down the coefficients of f(x): 6, -8, -5, -6, and 6.

Place the zero of g(x), which is 6, in a box to the left of these coefficients.
Bring down the first coefficient (6) to the bottom row.
Multiply the number in the box (6) by the number just brought down (6) and put this number under the next coefficient (-8).
Add the numbers in the second column and place the result in the bottom row. Repeat this process for the remaining coefficients.
The bottom row, except the last number, represents the coefficients of the quotient[tex]6x^3 + 28x^2 + 163x + 978.[/tex]
The last number in the bottom row is the remainder, -582.

Therefore, the quotient is [tex]6x^3 + 28x^2 + 163x + 978[/tex]and the remainder is -582, which corresponds to option (a).