To factor the expression \( x^3 - 5x^2 + 6x \), we can start by factoring out the greatest common factor (GCF). In this case, the GCF is \( x \):
\[ x(x^2 - 5x + 6) \]
Next, we can factor the quadratic expression \( x^2 - 5x + 6 \). We need to find two numbers that multiply to \( 6 \) (the constant term) and add up to \( -5 \) (the coefficient of the linear term). The numbers \( -2 \) and \( -3 \) fit these criteria.
Thus, we can factor the quadratic as follows:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]
Putting it all together, we have:
\[ x^3 - 5x^2 + 6x = x(x - 2)(x - 3) \]
So the factored form of the expression is:
\[ \boxed{x(x - 2)(x - 3)} \]