c) \( \left(2 x^{4}-6 x^{3}+3 x^{2}-1\right):\left(x^{2}-3 x+2\right) \)

To tackle the polynomial division \( \left(2 x^{4}-6 x^{3}+3 x^{2}-1\right):\left(x^{2}-3 x+2\right) \), we want to first factor the divisor. The expression \( x^{2} - 3x + 2 \) factors to \( (x-1)(x-2) \). Now, we'll perform polynomial long division where we divide \( 2x^4 - 6x^3 + 3x^2 - 1 \) by \( x^2 - 3x + 2 \). The division proceeds by determining how many times \( x^2 \) goes into \( 2x^4 \) (which is \( 2x^2 \)), then multiplying and subtracting from the original polynomial. Continue this process to find the quotient and remainder. As you simplify, be thorough with each step and combine like terms as you go along, ultimately arriving either at a quotient with a possible remainder or realizing further simplifications. The resultant quotient will likely be a polynomial in \( x \) with a degree lower than that of the original numerator. Enjoy the process; it’s like unwrapping a gift to see what’s inside!