Factor polynomal \( b^{2}-2 b^{2}+3 b-6 \)

To factor the polynomial \( b^{2}-2 b^{2}+3 b-6 \), first combine like terms. The expression simplifies to \( -b^{2}+3 b-6 \), or you can factor out a negative sign to get \( -(b^{2}-3b+6) \). However, this quadratic does not factor neatly with rational numbers. Therefore, we can use the quadratic formula to find the roots instead: \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \) where \( A = -1, B = 3, \) and \( C = -6 \). This helps find factors if necessary! To rewrite it in factored form, you can express it as \( -(b - 3)^2 + 3 \), which shows the vertex form of the polynomial, but does not factor over the rationals. Remember, not all polynomials can be factored in simple forms!