Step 1 :The remainder theorem states that if a polynomial \(P(x)\) is divided by \((x-a)\), then the remainder is \(P(a)\).
Step 2 :In this case, we are asked to find \(P(-2)\), so we substitute -2 into the polynomial:
Step 3 :\(P(-2) = (-2)^3 + 2*(-2)^2 + 7 = -8 + 8 + 7 = 7\).
Step 4 :So, \(P(-2) = 7\).
Step 5 :However, the question also asks for the quotient and the remainder of the associated division. Since we are evaluating \(P(-2)\), we are effectively dividing \(P(x)\) by \((x - (-2))\), or \((x + 2)\).
Step 6 :But since the value we found, \(P(-2)\), is not zero, this means that \((x + 2)\) is not a factor of \(P(x)\), and the division does not result in a polynomial with a degree less than \(P(x)\).
Step 7 :Therefore, the quotient of the division is the same as the original polynomial \(P(x)\), and the remainder is zero.
Step 8 :So, the quotient is \(x^3 + 2x^2 + 7\), the remainder is 0, and \(P(-2) = 7\).
Step 9 :The final answer is: \(\boxed{P(-2) = 7}\), the quotient is \(x^3 + 2x^2 + 7\), and the remainder is \(0\).