To determine the value of the discriminant for the given quadratic equation [tex]\(0 = x + 2 + x^2\)[/tex], we need to follow these steps:
1. Rewrite the equation in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[
x^2 + x + 2 = 0
\][/tex]
2. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
- [tex]\(a\)[/tex] is the coefficient of [tex]\(x^2\)[/tex], which is 1.
- [tex]\(b\)[/tex] is the coefficient of [tex]\(x\)[/tex], which is 1.
- [tex]\(c\)[/tex] is the constant term, which is 2.
3. Apply the formula for the discriminant, which is [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
4. Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[
\text{Discriminant} = 1^2 - 4 \cdot 1 \cdot 2
\][/tex]
5. Calculate the expression:
[tex]\[
\text{Discriminant} = 1 - 8
\][/tex]
[tex]\[
\text{Discriminant} = -7
\][/tex]
So, the value of the discriminant for the quadratic equation [tex]\(0 = x + 2 + x^2\)[/tex] is [tex]\(-7\)[/tex].
Therefore, the correct answer is [tex]\(-7\)[/tex].
