To find the highest degree of the expression \((3x^2 + 4)(x^5 - 3) - 12x^9 - 24x^3 + 2\), we need to determine the degree of each term and then find the highest one.
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First term: \((3x^2 + 4)(x^5 - 3)\)
- The degree of \(3x^2\) is 2 and the degree of \(x^5\) is 5.
- When we multiply these, the highest degree will be \(2 + 5 = 7\).
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Second term: \(-12x^9\)
- The degree is \(9\).
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Third term: \(-24x^3\)
- The degree is \(3\).
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Fourth term: \(+2\)
- The degree is \(0\) since it is a constant term.
Now, we compare the highest degrees from the terms:
- From \((3x^2 + 4)(x^5 - 3)\), the highest degree is \(7\).
- From \(-12x^9\), the highest degree is \(9\).
- From \(-24x^3\), the highest degree is \(3\).
- From \(+2\), the highest degree is \(0\).
The highest degree among these is \(9\).
Therefore, the highest degree for the overall expression is \(9\).
The answer is 9.