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$$\sqrt{(x + 1)(x + 2)(x + 3)(x + 4) + 1} =$$
$$\pm(x^2 + 5x + 4)$$
$$\pm(x^2 + 5x + 5)$$
$$\pm(x^2 + 5x + 6)$$
$$\pm(x^2 + 6x + 5)$$
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