To simplify the expression \(3x^2 + 2x - 5x - 2x^2 + 2\), we first combine like terms.
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Combine the terms involving \(x^2\): \[ 3x^2 - 2x^2 = x^2 \]
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Combine the terms involving \(x\): \[ 2x - 5x = -3x \]
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Combine the constant terms: \[ 2 \]
Putting it all together, we have: \[ x^2 - 3x + 2 \]
Next, we will factor the quadratic \(x^2 - 3x + 2\). To factor it, we need to find two numbers that multiply to \(2\) (the constant term) and add to \(-3\) (the coefficient of \(x\)).
The numbers that satisfy these conditions are \(-1\) and \(-2\), since: \[ -1 \cdot -2 = 2 \quad \text{and} \quad -1 + -2 = -3 \]
Thus, we can factor the expression as: \[ (x - 1)(x - 2) \]
Now, let's compare our factored result with the provided options:
- A) \((x + 1)(x + 2)\)
- B) \((x - 2)(x - 1)\)
- C) \((x + 2)(x - 2)\)
- D) \((3x + 2)(x + 1)\)
The correct option that matches our factorization \( (x - 1)(x - 2) \) is:
B) \((x - 2)(x - 1)\)
Thus, the answer is B.