the remainder theorem to find $P(-2)$ for $P(x)=x^{3}+2 x^{2}+7$.
ecifically, give the quotient and the remainder for the associated division and the value of $P(-2)$.
\[
\text { Quotient }=\square
\]
\[
\text { Remainder }=*
\]
\[
P(-2)=\square
\]

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 \(0\).