Factor \( -2 b k^{2}+6 b k-2 b \) \( -2 b\left(k^{2}-3 k+1\right) \) \( -2 b\left(k^{2}-3 k-1\right) \) \( -2 b\left(k^{2}+3 k+1\right) \)

To factor the expression \( -2 b k^{2}+6 b k-2 b \), start by factoring out the common term \(-2b\), which gives us: \[ -2b(k^2 - 3k + 1) \] Next, to find the correct option, we’ll check if \(k^2 - 3k + 1\) can be factored or solved. The discriminant \(\Delta = (-3)^2 - 4(1)(1) = 9 - 4 = 5\) implies that this quadratic does not factor neatly over the integers. Thus, the only choice that matches our factored form is: \(-2b(k^2 - 3k + 1)\). Now, let's explore some intriguing facts: The expression showcases the beauty of quadratic functions, where the solutions represent the x-intercepts of the parabola formed by plotting \(y = k^2 - 3k + 1\). It’s fascinating how symmetry plays a part in quadratic functions, with the vertex representing the minimum or maximum point, depending on the sign of the leading coefficient! Furthermore, there’s a handy trick for checking your factors—plugging in simple values for \(k\). By substituting values like \(k=0\) or \(k=1\), you can easily evaluate whether the original polynomial equals zero. It’s like a treasure hunt for the right factors!