We are asked to solve the following equation for $x$: $a(x - \frac{1}{a^2}) + a^2(x - \frac{1}{a}) = 2$.

1. Problem Description

We are asked to solve the following equation for $x$:

$a(x - \frac{1}{a^2}) + a^2(x - \frac{1}{a}) = 2$.

2. Solution Steps

First, we distribute $a$ and $a^2$ into the parenthesis:

$ax - \frac{a}{a^2} + a^2x - \frac{a^2}{a} = 2$

Simplify the fractions:

$ax - \frac{1}{a} + a^2x - a = 2$

Now, we group the terms with $x$:

$ax + a^2x = 2 + a + \frac{1}{a}$

Factor out $x$:

$x(a + a^2) = 2 + a + \frac{1}{a}$

$x(a(1+a)) = \frac{2a + a^2 + 1}{a}$

$x(a(1+a)) = \frac{a^2 + 2a + 1}{a}$

$x(a(1+a)) = \frac{(a+1)^2}{a}$

Divide both sides by $a(1+a)$:

$x = \frac{(a+1)^2}{a \cdot a(a+1)}$

$x = \frac{(a+1)^2}{a^2(a+1)}$

$x = \frac{a+1}{a^2}$

3. Final Answer

$x = \frac{a+1}{a^2}$