To expand the expression \((3x^2 - x + 3)(3x^2 + x - 2)\), we will use the distributive property (also known as the FOIL method for binomials). We'll multiply each term from the first polynomial by each term from the second polynomial.
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Distributing \(3x^2\):
- \(3x^2 \cdot 3x^2 = 9x^4\)
- \(3x^2 \cdot x = 3x^3\)
- \(3x^2 \cdot -2 = -6x^2\)
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Distributing \(-x\):
- \(-x \cdot 3x^2 = -3x^3\)
- \(-x \cdot x = -x^2\)
- \(-x \cdot -2 = 2x\)
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Distributing \(3\):
- \(3 \cdot 3x^2 = 9x^2\)
- \(3 \cdot x = 3x\)
- \(3 \cdot -2 = -6\)
Now, we will combine all these results:
\[ 9x^4 + (3x^3 - 3x^3) + (-6x^2 - x^2 + 9x^2) + (2x + 3x) - 6 \]
Next, we combine like terms:
- For \(x^4\) terms: \(9x^4\)
- For \(x^3\) terms: \(3x^3 - 3x^3 = 0\)
- For \(x^2\) terms: \(-6x^2 - x^2 + 9x^2 = 2x^2\)
- For \(x\) terms: \(2x + 3x = 5x\)
- The constant term: \(-6\)
Now we can write the expanded polynomial:
\[ 9x^4 + 2x^2 + 5x - 6 \]
So, the final answer in standard form is:
\[ \boxed{9x^4 + 2x^2 + 5x - 6} \]
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