To find the first step at which Karim makes an error, let's examine each of his steps.
The original quadratic equation is:
\[ x^2 + 5x - 24 = 0 \]
Step 1: \( x^2 + 8x + (-3x) - 24 = 0 \)
In this step, Karim is attempting to split the middle term \(5x\) into two parts. For his grouping to work, the two parts must sum to \(5x\) and their product must equal \(-24\). The term he chose, \(8x\) and \(-3x\), sums to \(5x\), which is correct, but let's ensure this was a legitimate split: \[ 8 + (-3) = 5 \] This part matches \(5x\), so this step is acceptable.
Step 2: \( (x^2 + 8x) + (-3x - 24) = 0 \)
This step is simply grouping the terms in a valid manner. There's no error here, as it represents a grouping of the quadratic terms and the constant.
Step 3: \( x(x + 8) + 3(x + 8) = 0 \)
Now, we factor out common factors from the grouped terms. The first group \(x^2 + 8x\) factors as \(x(x + 8)\). In the second group \(-3x - 24\) should actually factor as \(-3(x + 8)\), but let’s check: \[ -3x - 24 = -3(x + 8) \] This shows that the factorization done in this step is indeed correct as well.
Step 4: \( (x + 8)(x + 3) = 0 \)
In this step, we combine the two groups: \[ x(x + 8) + 3(x + 8) = (x + 8)(x + 3) = 0 \] This is also a valid step because the two terms that were factored were grouped properly.
Step 5: \( x + 8 = 0 \) or \( x + 3 = 0 \)
Karim proceeds to solve the factors correctly: \[ x + 8 = 0 \implies x = -8 \] and \[ x + 3 = 0 \implies x = -3 \]
Up to this point, the workings are consistent and valid.
The crux of the error comes from Step 1, where he incorrectly represented the break of \(5x\) into \(8x\) and \(-3x\). The correct factorization of the quadratic expression would instead utilize values that, when multiplied together, yield \(ac = -24\) (where \(a = 1\) and \(c = -24\)).
The correct pairs could be \(8\) and \(-3\) or incorrect as per the quadratic expression, leading the entire subsequent calculation into error. The grouping method Karim described does not lead to an incorrect approach per se, but logically, choosing non-factors (\(8\) and \(-3\)) was misleading for the original quadratic solution.
In the context of internal accuracy, the provided split at Step 1 is where Karim primarily faltered—it was misleading although mathematically valid if considered outside context without verifying. So, while not stepping into an outright algebraic failure, \(5x\) must yield a product breakdown conducive to the overall quadratic solution.
Thus, while all steps might seem valid, the initial step's choice is the source of the pursing trace errors.
Hence, the first error occurs at:
\[ \boxed{1} \]