To simplify the expression \(\frac{7x^2+2x-9}{7x+9}+\left(x+2\right)\left(x-3\right)\), we can start by simplifying each part separately.
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First, let's simplify the product \((x+2)(x-3)\): \[ (x+2)(x-3) = x^2 - 3x + 2x - 6 = x^2 - x - 6 \]
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Now, we rewrite the entire expression: \[ \frac{7x^2 + 2x - 9}{7x + 9} + x^2 - x - 6 \]
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Next, we need a common denominator to combine the terms. The common denominator is \(7x + 9\): \[ \frac{7x^2 + 2x - 9}{7x + 9} + \frac{(x^2 - x - 6)(7x + 9)}{7x + 9} \]
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We'll now expand \((x^2 - x - 6)(7x + 9)\): \[ (x^2 - x - 6)(7x + 9) = x^2(7x + 9) - x(7x + 9) - 6(7x + 9) \] This becomes: \[ 7x^3 + 9x^2 - 7x^2 - 9x - 42x - 54 = 7x^3 + (9x^2 - 7x^2) - (9x + 42x) - 54 \] Which simplifies to: \[ 7x^3 + 2x^2 - 51x - 54 \]
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Bringing everything together, we now have: \[ \frac{7x^2 + 2x - 9 + 7x^3 + 2x^2 - 51x - 54}{7x + 9} \]
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Combine all the terms in the numerator: \[ = \frac{7x^3 + (7x^2 + 2x^2) + (2x - 51x) + (-9 - 54)}{7x + 9} \] Which simplifies to: \[ = \frac{7x^3 + 9x^2 - 49x - 63}{7x + 9} \]
At this point, you may consider factoring the numerator further or finding its roots, but this is as far as we can simplify without a specific value for \(x\).
Thus, the final expression is: \[ \frac{7x^3 + 9x^2 - 49x - 63}{7x + 9} \]