Use the remainder theorem to find $$ P(-3) $$ for $$ P(x)=-2 x^{3}-5 x^{2}-4 $$. Specifically, give the quotient and the remainder for the associated division and the value of $$ P(-3) $$. Quotient $$ = $$ Remainder $$ =\square $$

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We need to divide
$$
P(x)=-2x^3-5x^2-4
$$
by $$x+3$$ to find the quotient and the remainder. Recall that by the Remainder Theorem, the remainder is $$P(-3)$$.

Since the polynomial is missing the $$x$$ term, we rewrite it as:
$$
-2x^3 - 5x^2 + 0x - 4.
$$

Perform synthetic division using $$c=-3$$:

1. Write the coefficients: $$-2,\;-5,\;0,\;-4$$.

2. Bring down the first coefficient:
   $$
   -2
   $$

3. Multiply $$-2$$ by $$-3$$ to get:
   $$
   (-2)(-3) = 6.
   $$

4. Add 6 to the next coefficient:
   $$
   -5 + 6 = 1.
   $$

5. Multiply $$1$$ by $$-3$$ to get:
   $$
   1 \cdot (-3) = -3.
   $$

6. Add $$-3$$ to the next coefficient:
   $$
   0 + (-3) = -3.
   $$

7. Multiply $$-3$$ by $$-3$$ to get:
   $$
   -3 \cdot (-3) = 9.
   $$

8. Add 9 to the constant term:
   $$
   -4 + 9 = 5.
   $$

The synthetic division yields the coefficients for the quotient and the remainder. The quotient corresponds to:
$$
-2x^2 + 1x - 3,
$$
and the remainder is:
$$
5.
$$

Thus, we have:
$$
\text{Quotient} = -2x^2 + x - 3,
$$
$$
\text{Remainder} = 5.
$$

By the Remainder Theorem, $$P(-3)$$ is equal to the remainder, so:
$$
P(-3)=5.
$$
Quotient: $$ -2x^2 + x - 3 $$ Remainder: $$ 5 $$ $$ P(-3) = 5 $$

Use the remainder theorem to find for .
Specifically, give the quotient and the remainder for the associated division and the value of .
Quotient
Remainder

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