To find an alternative method for computing the product \( 5 \frac{1}{4} \times \left(- \frac{11}{12}\right) \), we begin by converting \( 5 \frac{1}{4} \) into an improper fraction.
\[ 5 \frac{1}{4} = \frac{21}{4} \]
Now we need to compute:
\[ \frac{21}{4} \times \left(- \frac{11}{12}\right) \]
To assess each option given:
Option A: \[ (-1)(-5 \times \frac{11}{12}) + (-\frac{1}{4} \times \frac{11}{12}) \] This doesn't directly represent our original product, but let's evaluate whether it's mathematically correct but involves various operations similar to our original product.
Option B: \[ (-5 \frac{1}{4})(\frac{11}{12}) \] This is equivalent to the original product but does not provide a different computing method, as it just restates the product.
Option C: \[ -\frac{11}{12}(5 - \frac{1}{4}) = -\frac{11}{12} (\frac{20}{4} - \frac{1}{4}) = -\frac{11}{12} \times \frac{19}{4} \] This is a different way to break down \( 5 \frac{1}{4} \) but results in a different product since it simplifies to a different multiplication than the original.
Option D: This option represents breaking \( 5 \frac{1}{4} \) into two parts: \[ (-5)(-\frac{11}{12}) + (-\frac{1}{4})(-\frac{11}{12}) = -\frac{5 \times 11}{12} + \frac{1 \times 11}{48} = -\frac{55}{12} + \frac{11}{48} \] That is similarly a breakdown but ends up complicating the computation.
Among the options provided, only option B \( (-5 \frac{1}{4})(\frac{11}{12}) \) directly matches the original operation.
The most fitting restatement that offers a different angle while maintaining correctness in representation for computing the product: The best alternative expression in terms of rewriting without introducing unnecessary complexity would still be alluding to redefining its numerical components, as the expression specializes in the original concept. Thus, A can be argued as an expression manipulating elements toward computation, while B retains factual numerical data alignment and C changing the internal structure leads us to requirements of being cautious with interpretations.
Hence, the answer is the original method itself as represented via option B, maintaining the essence of the arithmetic operation.