1. **State the problem:** We are given the integral equation $$\int_0^{x^2} f(t) \, dt = x^2(1 + x)$$ and asked to find the value of $$f(4)$$. 2. **Recall the Fundamental Theorem of Calculus and chain rule:** If $$F(x) = \int_a^{g(x)} f(t) \, dt$$, then $$F'(x) = f(g(x)) \cdot g'(x)$$. 3. **Apply this to our problem:** Let $$F(x) = \int_0^{x^2} f(t) \, dt$$. Then $$F'(x) = f(x^2) \cdot \frac{d}{dx}(x^2) = f(x^2) \cdot 2x$$. 4. **Differentiate the right side:** Given $$F(x) = x^2(1 + x) = x^2 + x^3$$, so $$F'(x) = 2x + 3x^2$$. 5. **Set derivatives equal:** $$f(x^2) \cdot 2x = 2x + 3x^2$$. 6. **Solve for $$f(x^2)$$:** $$f(x^2) = \frac{2x + 3x^2}{2x}$$. 7. **Simplify the fraction:** $$f(x^2) = \frac{\cancel{2x} + 3x^2}{\cancel{2x}} = 1 + \frac{3x^2}{2x} = 1 + \frac{3x}{2}$$. 8. **Rewrite in terms of $$t = x^2$$:** Since $$t = x^2$$, then $$x = \sqrt{t}$$. So, $$f(t) = 1 + \frac{3}{2} \sqrt{t}$$. 9. **Find $$f(4)$$:** $$f(4) = 1 + \frac{3}{2} \sqrt{4} = 1 + \frac{3}{2} \times 2 = 1 + 3 = 4$$. **Final answer:** $$f(4) = 4$$.