To determine which of the given quadratic equations can be solved by grouping, we can look for quadratic equations that can be factored easily into two binomials.
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\(x^2 - 4x - 8 = 0\): This does not factor nicely using integers, as there are no two numbers that multiply to \(-8\) and add to \(-4\).
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\(x^2 + 10x + 21 = 0\): This factors nicely as \((x + 3)(x + 7) = 0\). It can be solved by factoring, but it does not require grouping.
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\(x^2 + 8x - 22 = 0\): To factor this using grouping, we look for two numbers that multiply to \(-22\) and add to \(8\). The numbers \(11\) and \(-2\) work because \(11 \cdot (-2) = -22\) and \(11 - 2 = 9\), so this quadratic can be rewritten and solved by grouping as \(x^2 + 11x - 2x - 22 = 0\).
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\(x^2 - 12x + 18 = 0\): This does not factor neatly using integers, as there are no two numbers that multiply to \(18\) and add to \(-12\).
Among the options, \(x^2 + 8x - 22 = 0\) can be solved by grouping after finding suitable pairs for factoring, making it the appropriate choice for this method.
Therefore, the quadratic equation that can be solved by grouping is:
\(x^2 + 8x - 22 = 0\)