To find the roots of the equation $4x^2 - 2(a^2 + b[x + a^2 b^2]) = 0$ using the quadratic formula, let's first simplify and organize the equation.

1. Expand the given equation:

   $4x^2 - 2(a^2 + bx + a^2b^2) = 0$

2. Distribute and simplify:

   $4x^2 - 2a^2 - 2bx - 2a^2b^2 = 0$

3. Rewrite the equation:

   $4x^2 - 2bx - 2a^2 - 2a^2b^2 = 0$

Now, rearrange the equation in standard quadratic form $ax^2 + bx + c = 0$:

$4x^2 - 2bx - 2(a^2 + a^2b^2) = 0$

Here, $a = 4$, $b = -2b$, and $c = -2(a^2 + a^2b^2)$.

4. Use the quadratic formula:

   $x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$

   where $A = 4$, $B = -2b$, and $C = -2(a^2 + a^2b^2)$.

Let's calculate the roots:

$x = \frac{-(-2b) \pm \sqrt{(-2b)^2 - 4(4)(-2(a^2 + a^2b^2))}}{2(4)}$

$x = \frac{2b \pm \sqrt{4b^2 + 32(a^2 + a^2b^2)}}{8}$

$x = \frac{b \pm \sqrt{b^2 + 8(a^2 + a^2b^2)}}{4}$

Thus, the roots of the given quadratic equation are:

$x = \frac{b + \sqrt{b^2 + 8(a^2 + a^2b^2)}}{4}$

$x = \frac{b - \sqrt{b^2 + 8(a^2 + a^2b^2)}}{4}$

Would you like further details or have any questions about the solution?
Here are some follow-up questions you might find interesting:

1. What does the discriminant tell us about the nature of the roots?
2. How can we factorize the given quadratic equation?
3. What are the vertex and axis of symmetry for this quadratic equation?
4. How does the value of $b$ affect the roots of the equation?
5. Can we solve the equation using a different method besides the quadratic formula?
6. What happens to the roots if $a$ or $b$ are zero?
7. How would the roots change if we modify the coefficient of $x^2$?
8. How can we graph the quadratic equation to visualize the roots?

Tip: Always check the discriminant ($B^2 - 4AC$) before solving quadratic equations to determine the nature of the roots (real and distinct, real and equal, or complex).