Analysis of the Inequality
To solve the given inequality 4^x

Solution Approach
- Start by rewriting the inequality as 4^x ≤ 3.2^x + √x + 4 + √x.
- Notice that 4^x can be written as (2^2)^x = 2^2x = (2^x)^2.
- Now rewrite the inequality as (2^x)^2 ≤ 3.2^x + √x + 4 + √x.

Further Simplification
- Let y = 2^x. The inequality now becomes y^2 ≤ 3(2y) + √(log2(y)) + 4 + √(log2(y)).
- This simplifies to y^2 - 6y - 4 - √(log2(y)) - √(log2(y)) ≤ 0.

Finding the Solution Set
- To find the solution set, we can analyze the behavior of the function y^2 - 6y - 4 - √(log2(y)) - √(log2(y)) and find where it is less than or equal to zero.
- By solving this inequality, we can determine the values of y (or 2^x) that satisfy the original inequality 4^x ≤ 3.2^x + √x + 4 + √x.

Conclusion
- The solution set of the given inequality will be the values of x corresponding to the values of y (2^x) that satisfy the inequality y^2 - 6y - 4 - √(log2(y)) - √(log2(y)) ≤ 0.
- Once the solution set is determined, it can be expressed in interval notation to represent the complete solution set of the inequality.