To multiply the polynomials \((x^4 + 7)(5x^2 + 10x - 1)\), we can use the distributive property, commonly referred to as the FOIL method when dealing with two binomials, but in this case, we're multiplying a polynomial of degree 4 by a polynomial of degree 2.
Here are the steps to multiply these two polynomials:
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Distribute each term in the first polynomial by each term in the second polynomial: \[ (x^4 + 7)(5x^2 + 10x - 1) = x^4(5x^2) + x^4(10x) + x^4(-1) + 7(5x^2) + 7(10x) + 7(-1) \]
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Perform the multiplications:
- \(x^4(5x^2) = 5x^6\)
- \(x^4(10x) = 10x^5\)
- \(x^4(-1) = -x^4\)
- \(7(5x^2) = 35x^2\)
- \(7(10x) = 70x\)
- \(7(-1) = -7\)
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Compile all the resulting terms: Combine all the terms together: \[ 5x^6 + 10x^5 - x^4 + 35x^2 + 70x - 7 \]
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Arrange the terms in descending order: Organizing the terms by their degree, we get: \[ 5x^6 + 10x^5 - x^4 + 70x + 35x^2 - 7 \]
So, the final product of the multiplication of the polynomials \((x^4 + 7)(5x^2 + 10x - 1)\) is: \[ \boxed{5x^6 + 10x^5 - x^4 + 35x^2 + 70x - 7} \]
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