5. which trinomial is equivalent to \( 3(x-2)^{2}-2(x-1) \)

To simplify the expression \( 3(x-2)^{2}-2(x-1) \), start by expanding \( (x-2)^{2} \) to get \( x^{2} - 4x + 4 \). So, multiplying by 3 gives \( 3(x^{2} - 4x + 4) = 3x^{2} - 12x + 12 \). Next, expand \( -2(x-1) \) to get \( -2x + 2 \). Combining both parts results in \( 3x^{2} - 12x + 12 - 2x + 2 = 3x^{2} - 14x + 14 \). Therefore, the equivalent trinomial is \( 3x^{2} - 14x + 14 \). Combining like terms can be tricky, and it’s easy to make mistakes! A common mistake is forgetting to distribute the coefficients correctly or to combine the constant terms during simplification. Always double-check each step, especially when expanding and simplifying. Keeping track of your signs is vital to avoid errors that could lead to a different result!