To solve the given expression using long division, we need to divide the polynomial [tex]\(18z^2 + 5z + 5\)[/tex] by the polynomial [tex]\(6z^2 - 4z + 1\)[/tex].
### Step-by-Step Long Division:
1. Divide the leading terms:
[tex]\[
\frac{18z^2}{6z^2} = 3
\][/tex]
So, the first term of the quotient is [tex]\(3\)[/tex].
2. Multiply the entire divisor by this term (3):
[tex]\[
3 \times (6z^2 - 4z + 1) = 18z^2 - 12z + 3
\][/tex]
3. Subtract this from the original polynomial:
[tex]\[
(18z^2 + 5z + 5) - (18z^2 - 12z + 3) = 17z + 2
\][/tex]
So, the quotient so far is [tex]\(3\)[/tex] and the remainder is [tex]\(17z + 2\)[/tex].
4. Express the remainder over the original divisor:
[tex]\[
\frac{17z + 2}{6z^2 - 4z + 1}
\][/tex]
### Final Answer:
Combining the quotient and the remainder, we get:
[tex]\[
\frac{18z^2 + 5z + 5}{6z^2 - 4z + 1} = 3 + \frac{17z + 2}{6z^2 - 4z + 1}
\][/tex]
So, the expression can be rewritten in the format [tex]\(q(x) + \frac{r(x)}{b(x)}\)[/tex] as follows:
[tex]\[
3 + \frac{17z + 2}{6z^2 - 4z + 1}
\][/tex]
This is the correct answer.
