Final answer:
The simplified form of the provided expression, considering variable restrictions, is (x - 2) where x ≠ 2, x ≠ 3, and x ≠ -7.
Step-by-step explanation:
The question is asking us to simplify the product of two rational expressions, (x2 - 49) / (x - 2) * (x2 - 5x + 6) / (x2 + 4x -21). Before simplifying, let's list out the restrictions on the variable x. x cannot be a value that makes the denominator of either fraction zero. So, x ≠ 2 from the first fraction and for the second fraction, by solving the quadratic equation x2 + 4x - 21 = 0, we obtain x ≠ 3, x ≠ -7. So, the restrictions on x are x ≠ 2, x ≠ 3, and x ≠ -7.
Now let's simplify the expressions. First, notice the first fraction, this can be factored as ((x - 7) * (x + 7)) / (x - 2). The second fraction can also be factored as (x - 2)(x - 3) / ((x - 3) * (x + 7)). When you multiply both expressions together, all terms cancel out except (x - 2) which is in the numerator. But remember, x cannot be 2, 3, or -7 due to restrictions. So, after considering restrictions, the simplified form of the given product is (x - 2), x ≠ 2, x ≠ 3, and x ≠ -7.
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