To find the highest degree of the expression \((3x^2 + 4)(x^5 - 3) - 12x^9 - 24x^3x + 2\), we will first expand the product and then combine the terms.
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Expand the product: \((3x^2 + 4)(x^5 - 3)\)
- The degree of \(3x^2\) is 2 and \(x^5\) is 5, so the highest degree term from this product is \(3x^2 \cdot x^5 = 3x^{7}\).
- The term \(4 \cdot x^5 = 4x^5\) is of degree 5.
- The term \(3x^2 \cdot (-3) = -9x^2\) is of degree 2.
- The term \(4 \cdot (-3) = -12\) is of degree 0.
Therefore, the highest degree term from the product is \(3x^7\) (degree 7).
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Other terms in the expression:
- The term \(-12x^9\) has degree 9.
- The term \(-24x^3x\) can be rewritten as \(-24x^4\) (degree 4).
- The constant \(+2\) has degree 0.
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Combine degrees:
- The highest degree from \((3x^2 + 4)(x^5 - 3)\) is 7.
- The highest degree from the other terms: \(-12x^9\) has degree 9.
Since the highest degree term in the entire expression comes from the \(-12x^9\), the overall highest degree of the given expression is:
\[ \boxed{9} \]