To find the highest degree of the expression \((3x^2 + 4)(x^5 - 3) - 12x^9 - \frac{24x}{3x + 2}\), we need to analyze each component of the expression separately and then combine the results.
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Calculate the degree of \((3x^2 + 4)(x^5 - 3)\):
- The degree of \(3x^2 + 4\) is 2 (the term \(3x^2\)).
- The degree of \(x^5 - 3\) is 5 (the term \(x^5\)).
- The degree of the product \((3x^2 + 4)(x^5 - 3)\) is \(2 + 5 = 7\).
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Calculate the degree of \(-12x^9\):
- The degree of \(-12x^9\) is 9 (the term \(-12x^9\)).
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Calculate the degree of \(-\frac{24x}{3x + 2}\):
- The degree of the numerator \(24x\) is 1.
- The degree of the denominator \(3x + 2\) is 1.
- The degree of the entire fraction \(-\frac{24x}{3x + 2}\) is \(1 - 1 = 0\) (the highest power in the numerator minus the highest power in the denominator).
Now, we combine the degrees of each part:
- The degree of \((3x^2 + 4)(x^5 - 3)\) is 7.
- The degree of \(-12x^9\) is 9.
- The degree of \(-\frac{24x}{3x + 2}\) is 0.
The highest degree from these components is clearly \(9\) (from \(-12x^9\)).
Thus, the highest degree for the entire expression \((3x^2 + 4)(x^5 - 3) - 12x^9 - \frac{24x}{3x + 2}\) is
\[ \boxed{9}. \]