To apply the Fundamental Theorem of Algebra, we first need to simplify the given equation:
\[ 12x - 6x^2 + 3x^4 = 6x^3 + 2x - x^4. \]
Next, let's move all terms to one side of the equation:
\[ 3x^4 + x^4 - 6x^3 - 6x^2 + 12x - 2x = 0. \]
Combining the like terms gives us:
\[ (3x^4 + x^4) - 6x^3 - 6x^2 + (12x - 2x) = 0, \] \[ 4x^4 - 6x^3 - 6x^2 + 10x = 0. \]
Now we can factor out the greatest common factor, which is 2x:
\[ 2x(2x^3 - 3x^2 - 3x + 5) = 0. \]
Now we can see that \(x = 0\) is one root. Next, we need to determine the number of roots of the cubic polynomial \(2x^3 - 3x^2 - 3x + 5\).
According to the Fundamental Theorem of Algebra, a polynomial of degree \(n\) has exactly \(n\) roots in the complex number system (counting multiplicities). The cubic polynomial \(2x^3 - 3x^2 - 3x + 5\) has a degree of 3.
Therefore, it will have a total of 3 roots (counting both real and complex roots).
Combining this with the root we factored out, the total number of roots for the original equation \(12x - 6x^2 + 3x^4 - (6x^3 + 2x - x^4) = 0\) is:
1 (from \(2x\)) + 3 (from \(2x^3 - 3x^2 - 3x + 5\)) = 4 roots.
Thus, the total number of roots for the original equation is 4.