To determine which quadratic equation can be solved by grouping, we look for a quadratic equation that can be factored into two binomials.
1. \(x^2 - 4x - 8 = 0\)
- Factors to find: we need two numbers that multiply to -8 and add to -4. There are no such integer pairs, so this does not factor nicely.
2. \(x^2 + 10x + 21 = 0\)
- Factors to find: we need two numbers that multiply to 21 and add to 10. \(3\) and \(7\) work. This equation factors to \((x + 3)(x + 7) = 0\). This can be solved by factoring, but not by grouping.
3. \(x^2 + 8x - 22 = 0\)
- Factors to find: we need two numbers that multiply to -22 and add to 8. The pairs do not lead to integer solutions that satisfy both conditions directly, so this cannot be factored easily without further manipulation.
4. \(x^2 - 12x + 18 = 0\)
- Factors to find: we need two numbers that multiply to 18 and add to -12. The pairs that work are \( -6 \) and \( -6 \) (this equals \( (x - 6)^2 = 0\) which is factorable).
While \(x^2 - 12x + 18\) can be factored fully, it does not specifically lend itself to grouping.
Conclusion: Most quadratics can be grouped effectively if they are structured to allow this. However, from the options presented, none clearly lend themselves to grouping method directly. Factoring is more suitable in general for all these equations.
None of these equations specifically demonstrate the need for grouping unless rearranged or manipulated further, but it would be best suited for \( x^2 + 8x - 22 = 0 \) if any were to be manipulated differently.
Thus, based on straightforward programming, the answer could either suggest \( x^2 + 8x - 22 = 0\) as it might approach grouping or get involved in manipulation routes.