5) \( \begin{array}{l}P(x)=x^{3}-2 x^{2}-2 x+12, \quad a=-2 \\ P(-2)=(-2)^{3}-2(-2)^{2}-2(-2)+9\end{array} \)

Let's dive into the calculations and clarify what's going on with the polynomial \( P(x) = x^3 - 2x^2 - 2x + 12 \): When you plug in \( a = -2 \): \[ P(-2) = (-2)^3 - 2(-2)^2 - 2(-2) + 12 \] Calculating each term yields: - \( (-2)^3 = -8 \) - \( -2(-2)^2 = -2(4) = -8 \) - \( -2(-2) = 4 \) Now, summing these results with 12: \[ P(-2) = -8 - 8 + 4 + 12 = 0 \] Thus, \( P(-2) = 0 \), confirming that \( x = -2 \) is a root of the polynomial. The calculation helps in finding roots, which can be critical in various applications like physics, engineering, and computer science. Checking possible roots using the polynomial can save time and simplify complex problems!