Answer: The correct option is (d) \(\textup{Step 4: }(x^2-9)(x+1);~\textup{Step 5: }(x-3)(x+3)(x+1).\)
Step-by-step explanation: The given steps in rewriting the expression \(x^3-9x+x^2-9\) as a product of three factors are as follows:
\(\textup{Step 1: }x^3-9x+x^2-9\\\\\textup{Step 2: }x^3+x^2-9x-9\\\\\textup{Step 3: }x^2(x+1)-9(x+1).\)
We are given to select the correct option that shows the next two steps to rewrite the expression.
We can factor the given expression as follows:
\(x^3-9x+x^2-9\\\\=x^3+x^2-9x-9\\\\=x^2(x+1)-9(x+1)\\\\=(x^2-9)(x+1)~~~~~~~~~~~\textup{[Step 4]}\\\\=(x-3)(x+3)(x+1).~~~~~~~\textup{[Step 5]}\)
Therefore, the next two steps are
\(\textup{Step 4: }(x^2-9)(x+1);~\textup{Step 5: }(x-3)(x+3)(x+1).\)
Thus, option (d) is corect.
