Use the remainder theorem to evaluate the polynomial for the given values of $$ x $$. $$ f(x)=2 x^{4}-9 x^{3}-4 x^{2}-5 x+10 $$

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**Step 1. Recall the Remainder Theorem**

The Remainder Theorem states that when a polynomial $$ f(x) $$ is divided by $$ (x - c) $$, the remainder is $$ f(c) $$.

**Step 2. Evaluate $$ f(x) $$ at a Given Value Using the Remainder Theorem**

If we wish to find the remainder when dividing $$ f(x) = 2x^4 - 9x^3 - 4x^2 - 5x + 10 $$ by $$ (x - c) $$, we just compute $$ f(c) $$.

For example, let’s evaluate the polynomial at two different values.

**Example 1. When $$ x = 2 $$**

Compute
$$
f(2) = 2(2)^4 - 9(2)^3 - 4(2)^2 - 5(2) + 10.
$$

Calculate each term:
- $$2(2)^4 = 2 \times 16 = 32,$$
- $$-9(2)^3 = -9 \times 8 = -72,$$
- $$-4(2)^2 = -4 \times 4 = -16,$$
- $$-5(2) = -10,$$
- Constant term: $$+10.$$

Now, combine the terms:
$$
32 - 72 - 16 - 10 + 10.
$$

Step-by-step sum:
- $$32 - 72 = -40,$$
- $$-40 - 16 = -56,$$
- $$-56 - 10 = -66,$$
- $$-66 + 10 = -56.$$

Thus, 
$$
f(2) = -56.
$$

**Example 2. When $$ x = -2 $$**

Compute
$$
f(-2) = 2(-2)^4 - 9(-2)^3 - 4(-2)^2 - 5(-2) + 10.
$$

Calculate each term:
- $$2(-2)^4 = 2 \times 16 = 32,$$
- $$-9(-2)^3 = -9 \times (-8) = 72,$$
- $$-4(-2)^2 = -4 \times 4 = -16,$$
- $$-5(-2) = 10,$$
- Constant term: $$+10.$$

Now, combine the terms:
$$
32 + 72 - 16 + 10 + 10.
$$

Step-by-step sum:
- $$32 + 72 = 104,$$
- $$104 - 16 = 88,$$
- $$88 + 10 = 98,$$
- $$98 + 10 = 108.$$

Thus,
$$
f(-2) = 108.
$$

**Final Answers:**

- The remainder when $$ f(x) $$ is divided by $$ (x - 2) $$ is $$ -56 $$ since $$ f(2) = -56 $$.
- The remainder when $$ f(x) $$ is divided by $$ (x + 2) $$ is $$ 108 $$ since $$ f(-2) = 108 $$.
When $$ x = 2 $$, the remainder is -56. When $$ x = -2 $$, the remainder is 108.

Use the remainder theorem to evaluate the polynomial for the given values of .

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