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Here is the problem: x squared plus 12x plus 27.

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Factor a polynomial over the rational numbers.

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Write the polynomial in standard form.

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Use the AC method. Multiply a times c equals 1 times 27 equals 27. Find two numbers whose product is 27 and whose sum is b equals 12: 9 and 3.

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Factor by grouping. Group the first two and last two terms, factor each group, then factor out the common binomial. The roots are negative 9 and negative 3.

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Write the complete factorization.

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So the answer is (x plus 3)(x plus 9).

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x squared plus 12x plus 27 factors completely over the rationals as (x plus 3)(x plus 9).

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Expanding (x plus 3)(x plus 9) gives back x squared plus 12x plus 27.