1. The problem is to find approximate solutions to the equation $$4^{x+2} + 2 = \frac{1}{3}x + 2$$ for $$x \in [-4,4]$$. 2. First, simplify the equation by subtracting 2 from both sides: $$4^{x+2} + 2 - 2 = \frac{1}{3}x + 2 - 2$$ which simplifies to $$4^{x+2} = \frac{1}{3}x$$ 3. This equation involves an exponential function on the left and a linear function on the right. Exact algebraic solutions are difficult, so we use the graph to estimate solutions. 4. From the graph, the exponential curve $$y = 4^{x+2} + 2$$ and the line $$y = \frac{1}{3}x + 2$$ intersect approximately at $$x = -2.1$$. 5. Checking the interval $$[-4,4]$$, there is no second intersection point visible on the graph, so the second solution does not exist in this range. 6. Therefore, the approximate solutions are: - First solution: $$x \approx -2.1$$ - Second solution: no solution in $$[-4,4]$$ Final answer: $$x \approx -2.1$$ only.