You want to find the value of x that satisfies the inequality:

a^2 + 2ax + x^2 > 0.25ax

First, simplify the expression on the left side of the inequality by factoring it:

(a + x)^2 > 0.25ax

Now, take the square root of both sides to solve for x:

|a + x| > 0.5√(ax)

Now, consider two cases:

If a > 0 and x > 0:
In this case, the original inequality simplifies to:
a^2 + 2ax + x^2 > 0.25ax
(a + x) > 0.5√(ax)

Therefore, the solution is x > 0.5√a.

If a < 0 and x > 0:
In this case, the original inequality simplifies to:
a^2 + 2ax + x^2 > 0.25ax
|a + x| > 0.5√(ax)

Therefore, the solution is x > max(0.5√(-a), -0.5√(-a)).

These are the solutions for x that satisfy the given inequality based on the two possible cases.