√(4х-х^2)>-2-3х^2

To solve the inequality \( \sqrt{4x - x^2} > -2 - 3x^2 \), we can break this down step by step.

1. First, simplify the inequality:
   We have a square root, which is always non-negative, so we can ignore the right side if it's negative. Also, we need to ensure that the expression inside the square root is non-negative.

   From the square root \( \sqrt{4x - x^2} \), the expression inside must be greater than or equal to zero:

   \[ 4x - x^2 \geq 0 \]

   Factoring gives:

   \[ x(4 - x) \geq 0 \]

   This inequality holds true for \( x \) in the intervals \( [0, 4] \).

2. Solve the inequality \( \sqrt{4x - x^2} > -2 - 3x^2 \):

   Since the left side (\( \sqrt{4x - x^2} \)) is always non-negative, we focus on the right side \( -2 - 3x^2 \). For the entire inequality to hold, we need:

   \[ \sqrt{4x - x^2} \geq 0 > -2 - 3x^2 \]

   This tells us that \( -2 - 3x^2 < 0 \). Solving this gives:

   \[ -3x^2 < 2 \]

   or

   \[ 3x^2 > -2 \]

   Since \( 3x^2 \) is always non-negative for real \( x \), this inequality is always satisfied for all real \( x \).

3. Combining the results: Therefore, we only need to consider the range from \( 0 \) to \( 4 \) where the square root is defined:

   \( x \in [0, 4] \)

4. Conclusion: The solution for the inequality \( \sqrt{4x - x^2} > -2 - 3x^2 \) is:

   \[ x \in [0, 4] \]

Thus, the solution to the inequality is \( x \in [0, 4] \).