Factor completely. \( 10 x^{3}-20 x^{2}-3 x+6 \)

To factor the polynomial \( 10 x^{3}-20 x^{2}-3 x+6 \) completely, we can start by grouping terms. Let's group the first two and the last two terms: \[ (10x^3 - 20x^2) + (-3x + 6). \] This allows us to factor out common factors from each group: \[ 10x^2(x - 2) - 3(x - 2). \] Now we can see that \( (x - 2) \) is a common factor: \[ (10x^2 - 3)(x - 2). \] Next, examine \( 10x^2 - 3 \) to see if it can be factored further. This expression does not factor nicely as it is a difference of squares only if we use complex numbers. Hence, the completely factored form is: \[ (10x^2 - 3)(x - 2). \]