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Here is the problem: 4x squared minus 12x minus 7.

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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 4 times negative 7 equals negative 28. Find two numbers whose product is negative 28 and whose sum is b equals negative 12: 2 and negative 14.

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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 1 over 2 and 7 over 2.

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Write the complete factorization.

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So the answer is (2x minus 7)(2x plus 1).

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4x squared minus 12x minus 7 factors completely over the rationals as (2x minus 7)(2x plus 1).

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Expanding (2x minus 7)(2x plus 1) gives back 4x squared minus 12x minus 7.