To find the product of the polynomials \((2x^2 + 4x^3)(3x^4 - 7x^3)\), we will use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial.
Let's perform the multiplication step by step:
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Multiply \(2x^2\) by \(3x^4\): \[ 2x^2 \cdot 3x^4 = 6x^{2+4} = 6x^6 \]
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Multiply \(2x^2\) by \(-7x^3\): \[ 2x^2 \cdot (-7x^3) = -14x^{2+3} = -14x^5 \]
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Multiply \(4x^3\) by \(3x^4\): \[ 4x^3 \cdot 3x^4 = 12x^{3+4} = 12x^7 \]
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Multiply \(4x^3\) by \(-7x^3\): \[ 4x^3 \cdot (-7x^3) = -28x^{3+3} = -28x^6 \]
Now we combine all the products:
- From steps 1 and 4: \(6x^6\) and \(-28x^6\) \[ 6x^6 - 28x^6 = -22x^6 \]
- From step 2: \(-14x^5\)
- From step 3: \(12x^7\)
Now we can write the complete product: \[ 12x^7 - 22x^6 - 14x^5 \]
Finally, we arrange the polynomial in standard form (descending order of exponents): \[ \boxed{12x^7 - 22x^6 - 14x^5} \]