1. The problem asks to find the Laplace transform of the function $$f(t) = 4t^3 + 7t^3 e^{12t}$$. 2. Recall the Laplace transform formulas: - For $$t^n$$, $$\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}$$. - For $$t^n e^{at}$$, $$\mathcal{L}\{t^n e^{at}\} = \frac{n!}{(s - a)^{n+1}}$$. 3. Calculate the Laplace transform of each term separately: - For $$4t^3$$: $$4 \times \frac{3!}{s^{4}} = 4 \times \frac{6}{s^{4}} = \frac{24}{s^{4}}$$. - For $$7t^3 e^{12t}$$: $$7 \times \frac{3!}{(s - 12)^{4}} = 7 \times \frac{6}{(s - 12)^{4}} = \frac{42}{(s - 12)^{4}}$$. 4. Combine the results: $$F(s) = \frac{24}{s^{4}} + \frac{42}{(s - 12)^{4}}$$. 5. None of the given options match this result, so none of the options a, b, c, or d correctly represent the Laplace transform of $$f(t)$$.