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Bot GPT 4-o mini To determine which expression simplifies to the same as \( 4x^2 - 3x^2 + 5x - 24 \), we first simplify the given expression:
\[
4x^2 - 3x^2 + 5x - 24 = (4 - 3)x^2 + 5x - 24 = 1x^2 + 5x - 24 = x^2 + 5x - 24.
\]
Next, we will factor this expression \( x^2 + 5x - 24 \).
To factor \( x^2 + 5x - 24 \), we are looking for two numbers that multiply to \(-24\) (the constant term) and add to \(5\) (the coefficient of \(x\)). The numbers \(8\) and \(-3\) satisfy these conditions because:
- \(8 \cdot (-3) = -24\)
- \(8 + (-3) = 5\)
Thus, we can factor the expression as:
\[
x^2 + 5x - 24 = (x + 8)(x - 3).
\]
Now, let's review the given options:
1. \( (x + 8)(x - 3) \) - This matches our factorization.
2. \( 4x(x - 6) \) - This does not match.
3. \( 2(x - 12) \) - This does not match.
4. \( (x + 6)(x - 4) \) - This does not match.
The expression that simplifies to the same expression as \( 4x^2 - 3x^2 + 5x - 24 \) is:
\[
\boxed{(x + 8)(x - 3)}.
\]