Factor by grouping: \( 6 x^{2}-12 x+3 x-6 \)

To factor by grouping, we first rearrange the terms: \( (6x^2 - 12x) + (3x - 6) \). Now, let's factor out the common factors from each group: From the first group \( 6x^2 - 12x \), we can factor out \( 6x \) to get: \( 6x(x - 2) \). From the second group \( 3x - 6 \), we can factor out \( 3 \) to get: \( 3(x - 2) \). Now we have: \( 6x(x - 2) + 3(x - 2) \). Notice that \( (x - 2) \) is a common factor. We factor that out: \( (x - 2)(6x + 3) \). Finally, we can simplify \( 6x + 3 \) by factoring out a \( 3 \): \( 3(2x + 1) \). So, the completely factored form of the original expression is: \( 3(x - 2)(2x + 1) \).