To simplify the expression \(4x^2 - 3x^2 + 5x - 24\), we first combine like terms.
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Combine \(4x^2\) and \(-3x^2\): \[ 4x^2 - 3x^2 = 1x^2 = x^2 \]
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We then write the expression: \[ x^2 + 5x - 24 \]
Now we want to factor the quadratic expression \(x^2 + 5x - 24\).
To factor \(x^2 + 5x - 24\), we look for two numbers that multiply to \(-24\) (the constant term) and add up to \(5\) (the coefficient of \(x\)). The numbers \(8\) and \(-3\) meet these criteria because:
- \(8 \times (-3) = -24\)
- \(8 + (-3) = 5\)
Thus, we can factor the expression as: \[ (x + 8)(x - 3) \]
We can check the other options provided to determine which one simplifies to the same expression as \(4x^2 - 3x^2 + 5x - 24\):
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\(4x(x-6)\): This expands to \(4x^2 - 24x\), which is not equivalent to \(x^2 + 5x - 24\).
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\((x+8)(x-3)\): We found that this is the factored form of the quadratic \(x^2 + 5x - 24\).
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\((x+6)(x-4)\): This expands to \(x^2 - 4x + 6x - 24 = x^2 + 2x - 24\), which is not equivalent.
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\(2(x-12)\): This expands to \(2x - 24\), which is not equivalent.
Thus, the correct answer is: \[ \text{(x+8)(x−3)} \]