To solve the equation:

$4x^6 - 2 = (x^3 + 1)^2 - 2x^3$

we’ll simplify and solve for $x$.

Step 1: Expand $(x^3 + 1)^2$

$(x^3 + 1)^2 = x^6 + 2x^3 + 1$

So, substituting this into the equation, we have:

$4x^6 - 2 = x^6 + 2x^3 + 1 - 2x^3$

Step 2: Simplify the Right Side

Notice that $+2x^3$ and $-2x^3$ cancel each other out, leaving:

$4x^6 - 2 = x^6 + 1$

Step 3: Rearrange Terms

Now, bring all terms involving $x$ to one side:

$4x^6 - x^6 = 1 + 2$

$3x^6 = 3$

Step 4: Divide by 3

$x^6 = 1$

Step 5: Solve for $x$

To solve for $x$, take the sixth root of both sides:

$x = \pm 1$

Final Answer

The solutions to the equation are:

$x = 1 \quad \text{and} \quad x = -1$

Would you like further details on any step, or have any questions?

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Here are 5 related questions to expand on this:

1. How can we solve higher-order polynomials with roots other than integers?
2. How would the solution change if we added additional terms to the equation?
3. What are the implications if this equation were an inequality instead?
4. How would you solve if the exponent were odd (e.g., $x^5$ instead of $x^6$)?
5. What techniques help simplify similar equations with multiple powers?

Tip: Always check for opportunities to factor or cancel terms; this often simplifies complex polynomial equations.