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Here is the graph of the function \( y(x) = 4e^{2x} + 3x^2 + 5 \). The red point indicates the intersection with the y-axis at \( (0, 9) \), which matches the initial condition given in the problem.

Would you like any additional details or have any questions about the solution? Here are some related questions to consider:

1. How would the graph change if the initial condition were different?
2. What is the effect of modifying the exponential term in the function?
3. How can we verify the slope at different points on the graph?
4. What does the concavity of this function tell us about its behavior?
5. How would we solve a similar problem with a different differential equation?

**Tip:** When solving differential equations, always remember to apply initial conditions to determine the specific solution out of the general family of solutions.

Learn how to graph the function y(x) = 4e^(2x) + 3x^2 + 5 with a detailed plot and analysis. This problem involves exponential and quadratic functions, providing insights into their intersections and behaviors. Suitable for advanced high school students.

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