To find the number of roots for the equation \( 12x - 6x^2 + 3x^4 = 6x^3 + 2r - x^4 \), we start by rearranging the equation to bring all terms to one side:
\[ 3x^4 + x^4 - 6x^3 - 6x^2 + 12x - 2r = 0 \]
Combining like terms, we get:
\[ 4x^4 - 6x^3 - 6x^2 + 12x - 2r = 0 \]
This is a polynomial equation of degree 4, as the highest power of \(x\) is 4 (the coefficient of \(x^4\) is 4).
The Fundamental Theorem of Algebra states that a polynomial of degree \(n\) has exactly \(n\) roots (counting multiplicities and considering complex roots) in the complex number system.
Since our polynomial is of degree 4, we can conclude:
- There are a total of 4 roots.
Thus, the number of roots for the equation is:
\[ \boxed{4} \]