To determine which of the given options is a quadratic inequality, let's review the characteristics of each:
1. Quadratic inequality: A quadratic inequality involves a quadratic expression (a polynomial of degree 2) where the inequality symbol (>, <, ≥, ≤) is used instead of the equality symbol (=).
2. Option 1: \(x^2 + 5x + 10 > 25\)
- This is a quadratic expression, \(x^2 + 5x + 10\), set greater than 25.
- Since it involves a polynomial of degree \(2\) and uses the '>' inequality symbol, this is a quadratic inequality.
3. Option 2: \(5x^2 + 3x + 12 = 0\)
- This is a quadratic equation because it involves a polynomial of degree \(2\) and an equality symbol.
4. Option 3: \(x^3 + 12 < -3\)
- This is a cubic inequality because it involves a polynomial of degree \(3\) and an inequality symbol.
5. Option 4: \(3x + 7 > 12\)
- This is a linear inequality because it involves a linear polynomial (degree \(1\)) and an inequality symbol.
Thus, given the examination of each option, the one that fulfills the criteria of a quadratic inequality is:
[tex]\[ x^2 + 5x + 10 > 25 \][/tex]
Hence, the correct answer is:
Option 1: [tex]\( x^2 + 5x + 10 > 25 \)[/tex]
