To find the product of the polynomials \((2x^2 + 3x + 2)(5x^2 - 10x + 3)\) using the distributive property, we will multiply each term in the first polynomial by each term in the second polynomial.
Let's denote the first polynomial as \(A = 2x^2 + 3x + 2\) and the second polynomial as \(B = 5x^2 - 10x + 3\).
Step 1: Multiply each term in \(A\) by each term in \(B\)
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Multiplying \(2x^2\) by each term in \(B\):
- \(2x^2 \cdot 5x^2 = 10x^4\)
- \(2x^2 \cdot (-10x) = -20x^3\)
- \(2x^2 \cdot 3 = 6x^2\)
So, the result of multiplying \(2x^2\) by \(B\) is: \[ 10x^4 - 20x^3 + 6x^2 \]
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Multiplying \(3x\) by each term in \(B\):
- \(3x \cdot 5x^2 = 15x^3\)
- \(3x \cdot (-10x) = -30x^2\)
- \(3x \cdot 3 = 9x\)
So, the result of multiplying \(3x\) by \(B\) is: \[ 15x^3 - 30x^2 + 9x \]
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Multiplying \(2\) by each term in \(B\):
- \(2 \cdot 5x^2 = 10x^2\)
- \(2 \cdot (-10x) = -20x\)
- \(2 \cdot 3 = 6\)
So, the result of multiplying \(2\) by \(B\) is: \[ 10x^2 - 20x + 6 \]
Step 2: Combine all results
Now add all the results together:
\[ (10x^4 - 20x^3 + 6x^2) + (15x^3 - 30x^2 + 9x) + (10x^2 - 20x + 6) \]
Step 3: Group like terms
Now, let's group the like terms:
- For \(x^4\): \(10x^4\)
- For \(x^3\): \(-20x^3 + 15x^3 = -5x^3\)
- For \(x^2\): \(6x^2 - 30x^2 + 10x^2 = -14x^2\)
- For \(x\): \(9x - 20x = -11x\)
- Constant: \(6\)
Step 4: Write the final result
Putting it all together, we have: \[ 10x^4 - 5x^3 - 14x^2 - 11x + 6 \]
So, the product of \((2x^2 + 3x + 2)(5x^2 - 10x + 3)\) is: \[ \boxed{10x^4 - 5x^3 - 14x^2 - 11x + 6} \]