We need to use synthetic division to divide the given polynomials:
[tex]\[
w + 6 \longdiv{-4w^4 - 26w^3 - 12w^2 - 3w - 36}
\][/tex]
Firstly, confirm that the divisor is in the form [tex]\(w + c\)[/tex]. Here it is [tex]\(w + 6\)[/tex]. Notice that the corresponding root for this divisor is [tex]\(w = -6\)[/tex].
Next, write down the coefficients of the dividend polynomial:
[tex]\[
-4, -26, -12, -3, -36
\][/tex]
Here’s a step-by-step procedure for synthetic division:
1. Write down the root of the divisor, which is -6.
2. Set up the synthetic division using the coefficients of the dividend and the root of the divisor.
[tex]\[
\begin{array}{r|rrrrr}
-6 & -4 & -26 & -12 & -3 & -36 \\
\end{array}
\][/tex]
3. Bring down the leading coefficient, -4, directly below the line:
[tex]\[
\begin{array}{r|rrrrr}
-6 & -4 & -26 & -12 & -3 & -36 \\
& & -4 & & & \\
\end{array}
\][/tex]
4. Multiply -4 (the number just brought down) by the root (-6), and write the result below the next coefficient in the dividend.
[tex]\[
-4 \times -6 = 24
\][/tex]
[tex]\[
\begin{array}{r|rrrrr}
-6 & -4 & -26 & -12 & -3 & -36 \\
& & -4 & 24 & & \\
\end{array}
\][/tex]
5. Add this result (24) to the next coefficient (-26):
[tex]\[
-26 + 24 = -2
\][/tex]
[tex]\[
\begin{array}{r|rrrrr}
-6 & -4 & -26 & -12 & -3 & -36 \\
& & -4 & 24 & -2 & \\
\end{array}
\][/tex]
6. Repeat this process for each of the remaining coefficients:
[tex]\[
-2 \times -6 = 12
\][/tex]
[tex]\[
\begin{array}{r|rrrrr}
-6 & -4 & -26 & -12 & -3 & -36 \\
& & -4 & 24 & -2 & 12 \\
\end{array}
\][/tex]
[tex]\[
-12 + 12 = 0
\][/tex]
[tex]\[
\begin{array}{r|rrrrr}
-6 & -4 & -26 & -12 & -3 & -36 \\
& & -4 & 24 & -2 & 12 & 0 \\
\end{array}
\][/tex]
[tex]\[
0 \times -6 = 0
\][/tex]
[tex]\[
\begin{array}{r|rrrrr}
-6 & -4 & -26 & -12 & -3 & -36 \\
& & -4 & 24 & -2 & 12 & 0 \\
\end{array}
\][/tex]
[tex]\[
-3 + 0 = -3
\][/tex]
[tex]\[
\begin{array}{r|rrrrr}
-6 & -4 & -26 & -12 & -3 & -36 \\
& & -4 & 24 & -2 & 12 & 0 & -3 \\
\end{array}
\][/tex]
[tex]\[
-3 \times -6 = 18
\][/tex]
[tex]\[
\begin{array}{r|rrrrr}
-6 & -4 & -26 & -12 & -3 & -36 \\
& & -4 & 24 & -2 & 12 & 0 & -3 & 18\\
\end{array}
\][/tex]
[tex]\[
-36 + 18 = -18
\][/tex]
[tex]\[
\begin{array}{r|rrrrr}
-6 & -4 & -26 & -12 & -3 & -36 \\
& & -4 & 24 & -2 & 12 & 0 & -3 & 18 & -18\\
\end{array}
\][/tex]
The coefficients for the quotient polynomial are the numbers we get in the row after the line excluding the last number which is our remainder.
So, the quotient is:
[tex]\[
-4w^3 - 2w^2 + 0w - 3
\][/tex]
And the remainder is:
[tex]\[
-18
\][/tex]
Thus, the quotient and remainder of the division are:
[tex]\[
\boxed{(-4w^3 - 2w^2 + 0w - 3)}
\][/tex]
[tex]\[
\boxed{-18}
\][/tex]
