Let's go through Karim's steps one by one to identify any errors.
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In Step 1, Karim rewrites the equation \(x^2 + 5x - 24 = 0\) as \(x^2 + 8x + (-3x) - 24 = 0\). This step is incorrect because \(8x - 3x = 5x\) is indeed correct algebraically. It could be argued that he split the middle term incorrectly, as the correct factors of \(-24\) that add up to \(5\) should be \(8\) and \(-3\), resulting in \(5 = 8 + (-3)\) being right, but the grouping that follows is misleading.
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In Step 2, he groups the terms: \((x^2 + 8x) + (-3x - 24) = 0\). This grouping is also acceptable following his previous step.
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In Step 3, he factors out \(x\) from \(x^2 + 8x\) and \(-3\) from \(-3x - 24\): \(x(x + 8) + 3(x + 8) = 0\).
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In Step 4, he combines them: \((x + 8)(x + 3) = 0\). This step is incorrect because he did not correctly factor out \(-3\) in Step 3. It should have been \(3(x + 8)\) leading to an invalid grouping since this leads to a false statement.
Karim's first error occurs at Step 3, where he incorrectly included a factor of \(3\) that changes the expression due to the minus sign used earlier when factoring out \(-3x - 24\). The correct factorization should have been \(x(x + 8) - 3(x + 8) = 0\), leading to incorrect product factors.
In essence, the steps correctly identify that \(x\) does not properly lead to the factoring identified at \(x + 3\) due to the grouping not aligning effectively throughout the process.
Thus, Karim's first mistake occurs at Step 3.
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