To determine which ordered pair is included in the solution set of the given system of inequalities:
[tex]\[
\begin{array}{l}
y > x^2 + 3 \\
y < x^2 - 3x + 2
\end{array}
\][/tex]
we need to verify which of the given points:
[tex]\[
(-2, 8), \, (0, 2), \, (0, 4), \, (2, 2)
\][/tex]
satisfy both inequalities.
Let's check each point one by one.
### Point [tex]\((-2, 8)\)[/tex]:
1. Check the first inequality: [tex]\(y > x^2 + 3\)[/tex]
[tex]\[
8 > (-2)^2 + 3 \implies 8 > 4 + 3 = 7
\][/tex]
This is true.
2. Check the second inequality: [tex]\(y < x^2 - 3x + 2\)[/tex]
[tex]\[
8 < (-2)^2 - 3(-2) + 2 \implies 8 < 4 + 6 + 2 = 12
\][/tex]
This is also true.
Since [tex]\((-2, 8)\)[/tex] satisfies both inequalities, it is part of the solution set.
### Point [tex]\((0, 2)\)[/tex]:
1. Check the first inequality: [tex]\(y > x^2 + 3\)[/tex]
[tex]\[
2 > 0^2 + 3 \implies 2 > 3
\][/tex]
This is false.
As [tex]\((0, 2)\)[/tex] does not satisfy the first inequality, it cannot be part of the solution set.
### Point [tex]\((0, 4)\)[/tex]:
1. Check the first inequality: [tex]\(y > x^2 + 3\)[/tex]
[tex]\[
4 > 0^2 + 3 \implies 4 > 3
\][/tex]
This is true.
2. Check the second inequality: [tex]\(y < x^2 - 3x + 2\)[/tex]
[tex]\[
4 < 0^2 - 3 \cdot 0 + 2 \implies 4 < 2
\][/tex]
This is false.
As [tex]\((0, 4)\)[/tex] does not satisfy the second inequality, it cannot be part of the solution set.
### Point [tex]\((2, 2)\)[/tex]:
1. Check the first inequality: [tex]\(y > x^2 + 3\)[/tex]
[tex]\[
2 > 2^2 + 3 \implies 2 > 4 + 3 = 7
\][/tex]
This is false.
As [tex]\((2, 2)\)[/tex] does not satisfy the first inequality, it cannot be part of the solution set.
### Conclusion
After checking all the points, we see that the only ordered pair that satisfies both inequalities is [tex]\((-2, 8)\)[/tex]. Therefore, the ordered pair included in the solution set to the given system is:
[tex]\[
\boxed{(-2, 8)}
\][/tex]
