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How many integers lie in the range of f(x) = x(x + 2)(x + 4)(x + 6) + 7 for x in the interval [-4, 2]?

1. 27
2. 28
3. 25
4. 26

Correct answer: 28

Solution

Group as [x(x+6)][(x+2)(x+4)] = (x²+6x)(x²+6x+8). Let t = x²+6x. On x in [-4,2], t ranges from its vertex value to its endpoint maximum. Then f = t(t+8)+7 = t² + 8t + 7 = (t+4)² - 9. Evaluate over the t-range to get the min and max of f, then count integers between them inclusive.

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