Let's analyze Karim's steps to determine where he made an error.
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Step 1: \( x^2 + 8x + (-3x) - 24 = 0 \)
The original equation is \( x^2 + 5x - 24 = 0 \). The correct split for \( 5x \) using grouping would be \( 8x + (-3x) \), which is correct because \( 8 + (-3) = 5 \). So, this step is valid.
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Step 2: \( (x^2 + 8x) + (-3x - 24) = 0 \)
In this step, Karim has grouped his terms correctly. \( (x^2 + 8x) \) and \( (-3x - 24) \) are indeed correctly grouped. So, this step is also valid.
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Step 3: \( x(x + 8) + 3(x + 8) = 0 \)
Here, he factored out \( x \) from the first grouped term and \( 3 \) from the second. However, he factored \( -3 \) instead of \( 3 \) since he was extracting the factors from \( -3x - 24 \), which should have factored as \( -3(x + 8) \). So this step has an error.
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Step 4: \( (x + 8)(x + 3) = 0 \)
Since his previous step had an error, this step also ends up being incorrect.
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Step 5:
This step is based on the incorrect factored expression from the previous step and thus produces incorrect solutions.
In conclusion, Karim's first mistake occurs at Step 3.