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To factor the quadratic expression \(x^2 - 4x + 24\), we start by finding its roots using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For the given quadratic \(x^2 - 4x + 24\): - \(a = 1\) - \(b = -4\) - \(c = 24\) Plugging these values into the quadratic formula: \[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 24}}{2 \cdot 1} \] \[ x = \frac{4 \pm \sqrt{16 - 96}}{2} \] \[ x = \frac{4 \pm \sqrt{-80}}{2} \] \[ x = \frac{4 \pm \sqrt{-(16 \cdot 5)}}{2} \] \[ x = \frac{4 \pm 4i\sqrt{5}}{2} \] \[ x = 2 \pm 2i\sqrt{5} \] Since the roots \(2 + 2i\sqrt{5}\) and \(2 - 2i\sqrt{5}\) involve imaginary numbers, the quadratic expression does not factor easily over the reals. However, we can express it in terms of its factors involving complex numbers: \[ x^2 - 4x + 24 = (x - (2 + 2i\sqrt{5}))(x - (2 - 2i\sqrt{5})) \] While this is the factorization involving complex roots, it cannot be factored further into simpler polynomial expressions with real coefficients. So, over the real numbers, the quadratic expression \(x^2 - 4x + 24\) does not factor any further.

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