To multiply the polynomials \((5x^2 - 3x + 10)(2x^2 + 4x - 2)\), we will use the distributive property (also known as the FOIL method for binomials). We will multiply each term in the first polynomial by each term in the second polynomial.
Let's break it down:
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First, multiply \(5x^2\) with each term in the second polynomial:
- \(5x^2 \cdot 2x^2 = 10x^4\)
- \(5x^2 \cdot 4x = 20x^3\)
- \(5x^2 \cdot (-2) = -10x^2\)
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Next, multiply \(-3x\) with each term in the second polynomial:
- \(-3x \cdot 2x^2 = -6x^3\)
- \(-3x \cdot 4x = -12x^2\)
- \(-3x \cdot (-2) = 6x\)
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Finally, multiply \(10\) with each term in the second polynomial:
- \(10 \cdot 2x^2 = 20x^2\)
- \(10 \cdot 4x = 40x\)
- \(10 \cdot (-2) = -20\)
Now, let's combine all these results:
\[ 10x^4 + 20x^3 - 10x^2 - 6x^3 - 12x^2 + 6x + 20x^2 + 40x - 20 \]
Next, we combine like terms:
- \(10x^4\) (there's only one \(x^4\) term)
- \(20x^3 - 6x^3 = 14x^3\)
- \(-10x^2 - 12x^2 + 20x^2 = -2x^2\)
- \(6x + 40x = 46x\)
- \(-20\) (there's only one constant term)
Putting it all together, we get:
\[ 10x^4 + 14x^3 - 2x^2 + 46x - 20 \]
Thus, the product of \((5x^2 - 3x + 10)(2x^2 + 4x - 2)\) is:
\[ \boxed{10x^4 + 14x^3 - 2x^2 + 46x - 20} \]