To find the product of the two polynomials \( (3x^2 - 4x)(9x^4 + 12x^3 + 16x^2) \), we will distributively multiply each term in the first polynomial by each term in the second polynomial.
Starting with the first polynomial \( 3x^2 - 4x \):
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Multiply \( 3x^2 \) by each term in \( 9x^4 + 12x^3 + 16x^2 \):
- \( 3x^2 \cdot 9x^4 = 27x^6 \)
- \( 3x^2 \cdot 12x^3 = 36x^5 \)
- \( 3x^2 \cdot 16x^2 = 48x^4 \)
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Multiply \( -4x \) by each term in \( 9x^4 + 12x^3 + 16x^2 \):
- \( -4x \cdot 9x^4 = -36x^5 \)
- \( -4x \cdot 12x^3 = -48x^4 \)
- \( -4x \cdot 16x^2 = -64x^3 \)
Now, combine all these terms:
\[ 27x^6 + 36x^5 + 48x^4 - 36x^5 - 48x^4 - 64x^3 \]
Next, we will combine like terms:
- The \( x^5 \) terms: \( 36x^5 - 36x^5 = 0 \)
- The \( x^4 \) terms: \( 48x^4 - 48x^4 = 0 \)
- The \( x^3 \) term: \( -64x^3 \) remains as is.
The remaining terms are:
\[ 27x^6 - 64x^3 \]
Thus, the product of the two polynomials is:
\[ \boxed{27x^6 - 64x^3} \]
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