Sure, let's walk through the steps to find the missing number in the synthetic division problem.
We start with the initial row of coefficients and the divisor used for synthetic division:
[tex]\[
\begin{array}{cccc}
2 & -2 & 3 & 4 \\
& 4 & 4 & 14 \\
\hline
\end{array}
\][/tex]
The first number of the bottom row is directly taken from the first number of the top row:
[tex]\[
\begin{array}{cccc}
2 & -2 & 3 & 4 \\
& 4 & 4 & 14 \\
\hline
2 & & & \\
\end{array}
\][/tex]
Next, multiply the 2 (from the bottom row) by the divisor (4) and write the result (8) below -2:
[tex]\[
\begin{array}{cccc}
2 & -2 & 3 & 4 \\
& 4 & 8 & 14 \\
\hline
2 & & & \\
\end{array}
\][/tex]
Add the numbers in the second column:
[tex]\[
\begin{array}{cccc}
2 & -2 & 3 & 4 \\
& 4 & 8 & 14 \\
\hline
2 & 2 & & \\
\end{array}
\][/tex]
Next, multiply the new value (2) by the divisor (4) and write the result (8) below 3:
[tex]\[
\begin{array}{cccc}
2 & -2 & 3 & 4 \\
& 4 & 8 & 8 \\
\hline
2 & 2 & & \\
\end{array}
\][/tex]
Add the numbers in the third column:
[tex]\[
\begin{array}{cccc}
2 & -2 & 3 & 4 \\
& 4 & 8 & 8 \\
\hline
2 & 2 & 11 & \\
\end{array}
\][/tex]
Now, compare the expected fourth term from the synthetic division setup [tex]\(4\)[/tex] added with [tex]\(14\)[/tex] :
[tex]\[
3 + 4 = 7
\][/tex]
Thus, the missing number required to keep the synthetic division consistent is:
[tex]\[
11 - 4 = 7
\][/tex]
So the correctly filled in row should be:
[tex]\[
\begin{array}{cccc}
2 & -2 & 3 & 4 \\
& 4 & 4 & 14 \\
\hline
2 & 2 & 7 & 18 \\
\end{array}
\][/tex]
Thus the missing number is [tex]\( \boxed{7} \)[/tex]. Answer choice D is correct.
