12) \( \left(3 v^{3}+2-4 v\right)-\left(3 v+2 v^{3}-4\right) \) A) \( -5 v^{3}-13 v+6 \) B) \( -5 v^{3}-7 v+6 \) C) \( -5 v^{3}-14 v+6 \) D) \( v^{3}-7 v+6 \)

To solve the expression \( \left(3 v^{3}+2-4 v\right)-\left(3 v+2 v^{3}-4\right) \), we first distribute the negative sign across the second set of parentheses: \[ 3 v^{3} + 2 - 4 v - (3 v + 2 v^{3} - 4) = 3 v^{3} + 2 - 4 v - 3 v - 2 v^{3} + 4 \] Next, we combine like terms: - The \( v^3 \) terms: \( 3 v^{3} - 2 v^{3} = v^{3} \) - The \( v \) terms: \( -4 v - 3 v = -7 v \) - The constant terms: \( 2 + 4 = 6 \) Putting it all together gives us: \[ v^{3} - 7 v + 6 \] The correct answer is therefore D) \( v^{3}-7 v+6 \). You crushed that problem! Now, let’s think about what a polynomial expression can do in real life. Polynomials like this can model various situations, from physics problems predicting projectile motion to economics estimating profit or loss. They help simplify complex relationships into manageable equations that can lead to better predictions and decisions. Also, have you ever wondered how polynomials can be manipulated? Consider this: careful attention to negative signs like we saw in your problem can prevent major errors. A common pitfall is not distributing the negative correctly, leading to wrong combinations of terms. When subtracting polynomials, always keep track of those pesky signs!