To find the highest degree of the expression \( (3x^2 + 4)(x^5 - 3) - 12x^9 - 24x^3 + 2 \), we need to calculate the degree of each term in the expression.
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First term: \( (3x^2 + 4)(x^5 - 3) \)
- The degree of \( 3x^2 \) is 2.
- The degree of \( x^5 \) is 5.
- The degree of the product \( (3x^2)(x^5) = 3x^{2+5} = 3x^7 \), which has a degree of 7.
- The constant term \( 4 \) multiplied by \( x^5 \) contributes \( 4x^5 \) which has a degree of 5.
- The total highest degree from \( (3x^2 + 4)(x^5 - 3) \) is therefore \( \max(7, 5) = 7 \).
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Second term: \( -12x^9 \)
- This term has a degree of 9.
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Third term: \( -24x^3 \)
- This term has a degree of 3.
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Fourth term: \( +2 \)
- This is a constant, so its degree is 0.
Now, we compare all the degrees we calculated:
- From \( (3x^2 + 4)(x^5 - 3) \), the maximum degree is 7.
- From \( -12x^9 \), the degree is 9.
- From \( -24x^3 \), the degree is 3.
- From \( +2 \), the degree is 0.
The highest degree among these is 9.
Thus, the highest degree of the entire expression is \( \boxed{9} \).