To expand the expression \((2x^2 + 3x + 2)(x^2 + 5x - 10)\), we will use the distributive property (also known as the FOIL method for binomials, but in this case, we have trinomials).
We will multiply each term in the first polynomial by each term in the second polynomial:
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Multiply \(2x^2\) by each term in \((x^2 + 5x - 10)\):
- \(2x^2 \cdot x^2 = 2x^4\)
- \(2x^2 \cdot 5x = 10x^3\)
- \(2x^2 \cdot (-10) = -20x^2\)
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Multiply \(3x\) by each term in \((x^2 + 5x - 10)\):
- \(3x \cdot x^2 = 3x^3\)
- \(3x \cdot 5x = 15x^2\)
- \(3x \cdot (-10) = -30x\)
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Multiply \(2\) by each term in \((x^2 + 5x - 10)\):
- \(2 \cdot x^2 = 2x^2\)
- \(2 \cdot 5x = 10x\)
- \(2 \cdot (-10) = -20\)
Now we combine all these results:
\[ 2x^4 + 10x^3 - 20x^2 + 3x^3 + 15x^2 - 30x + 2x^2 + 10x - 20 \]
Next, we combine like terms:
- Combine \(x^4\) terms: \(2x^4\)
- Combine \(x^3\) terms: \(10x^3 + 3x^3 = 13x^3\)
- Combine \(x^2\) terms: \(-20x^2 + 15x^2 + 2x^2 = -3x^2\)
- Combine \(x\) terms: \(-30x + 10x = -20x\)
- The constant term: \(-20\)
Putting it all together, we get the expanded expression in standard form:
\[ \boxed{2x^4 + 13x^3 - 3x^2 - 20x - 20} \]
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