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A factored polynomial reveals its roots, a key concept in understanding the behavior of these expressions.

Simplify \[\frac{x^2+11x+18}{x+2} \div \frac{x+9}{12}.\]

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Simplify \[\frac{4x^2+20x+21}{3x+6} \times \frac{2x+3}{2x+7} \div \frac{4x^2+12x+9}{6x+12}.\]

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If \[\frac{x}{x-7}-\frac{x}{x+9}=\frac{ax}{(x-7)(x+9)},\]

holds for all values of \(x\), what is the value of \(a\)?

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