To find the possible values for [tex]\( c \)[/tex], we need to compare coefficients from the given polynomial equation [tex]\((a x + 9)(b x + 2) = 27 x^2 + c x + 18\)[/tex].
Let's start by expanding the left-hand side of the given equation:
[tex]\[
(a x + 9)(b x + 2) = a b x^2 + 2 a x + 9 b x + 18
\][/tex]
Combining like terms, the polynomial becomes:
[tex]\[
a b x^2 + (2 a + 9 b) x + 18
\][/tex]
We can now compare coefficients from both sides of the equation:
[tex]\[
a b x^2 + (2 a + 9 b) x + 18 = 27 x^2 + c x + 18
\][/tex]
From comparing the coefficients, we obtain the following equations:
1. [tex]\(a b = 27\)[/tex]
2. [tex]\(2 a + 9 b = c\)[/tex]
3. The constant term 18 is the same on both sides, so no new information is required from it.
Additionally, we are given another condition:
[tex]\[
a + b = 12
\][/tex]
We now have a system of equations to solve:
[tex]\[
\begin{cases}
a b = 27 \\
a + b = 12
\end{cases}
\][/tex]
We can solve this system step-by-step:
1. From [tex]\(a + b = 12\)[/tex], express [tex]\( b \)[/tex] in terms of [tex]\( a \)[/tex]:
[tex]\[
b = 12 - a
\][/tex]
2. Substitute [tex]\( b = 12 - a \)[/tex] into [tex]\( a b = 27 \)[/tex]:
[tex]\[
a (12 - a) = 27
\][/tex]
This simplifies to:
[tex]\[
12a - a^2 = 27
\][/tex]
[tex]\[
a^2 - 12a + 27 = 0
\][/tex]
3. Solve this quadratic equation for [tex]\( a \)[/tex] using the quadratic formula [tex]\( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[
a = \frac{12 \pm \sqrt{144 - 108}}{2}
\][/tex]
[tex]\[
a = \frac{12 \pm \sqrt{36}}{2}
\][/tex]
[tex]\[
a = \frac{12 \pm 6}{2}
\][/tex]
This gives us two possible values for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{18}{2} = 9 \quad \text{or} \quad a = \frac{6}{2} = 3
\][/tex]
4. Calculate the corresponding values for [tex]\( b \)[/tex]:
[tex]\[
\text{If } a = 9, \text{ then } b = 12 - 9 = 3
\][/tex]
[tex]\[
\text{If } a = 3, \text{ then } b = 12 - 3 = 9
\][/tex]
5. Finally, substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into [tex]\( 2a + 9b = c \)[/tex] to find the possible values of [tex]\( c \)[/tex]:
[tex]\[
\text{If } a = 9 \text{ and } b = 3, \text{ then }
\][/tex]
[tex]\[
c = 2(9) + 9(3) = 18 + 27 = 45
\][/tex]
[tex]\[
\text{If } a = 3 \text{ and } b = 9, \text{ then }
\][/tex]
[tex]\[
c = 2(3) + 9(9) = 6 + 81 = 87
\][/tex]
Thus, the two possible values for [tex]\( c \)[/tex] are 45 and 87, which corresponds to option (C).
The correct answer is:
[tex]\[
\boxed{45, 87}
\][/tex]
