Final answer:
To solve P(x) = x³-5x² + 8x - 6 using long division and evaluate P(-2), divide each term by x + 2 and repeat the process until you reach the constant term. Add the constant term to the result to obtain the value of P(-2).
Step-by-step explanation:
To solve P(x) = x³-5x² + 8x - 6 using long division, we can use the synthetic division method. The divisor will be x + 2, since we want to evaluate P(-2). First, rewrite the polynomial in descending order, with placeholders for missing terms: x³ - 5x² + 8x - 6. Now, divide each term by the divisor and write the result in a horizontal line. Start with the first term, x³, and divide it by x, which gives x². Then, multiply the divisor x + 2 by x² to get x³ + 2x². Subtract this from the original polynomial to get -7x² + 8x - 6. Repeat the process with this new polynomial, dividing each term by x and multiplying the divisor by the result. Continue until you reach the constant term. Finally, add the constant term to the result. The final result will give you the value of P(-2).