Solve the whole problem and find the quotient and remainder. 

Let's continue solving the entire problem to find both the quotient and the remainder.

### The Problem

We need to divide:

\[
\frac{x^5 + 3x^4 - 5x^3 + 14x^2 + 30x - 16}{x^3 - x^2 + x + 8}
\]

### Step 1: First Division

Start by dividing the first term of the numerator by the first term of the denominator:

\[
\frac{x^5}{x^3} = x^{2}
\]

**Quotient so far: \(x^2\)**

Now, multiply \(x^2\) by the entire denominator:

\[
x^2 \times (x^3 - x^2 + x + 8) = x^5 - x^4 + x^3 + 8x^2
\]

Subtract this from the original numerator:

\[
(x^5 + 3x^4 - 5x^3 + 14x^2 + 30x - 16) - (x^5 - x^4 + x^3 + 8x^2)
\]

This gives:

\[
(3x^4 + x^4) + (-5x^3 - x^3) + (14x^2 - 8x^2) + 30x - 16
\]

Simplifying:

\[
4x^4 - 6x^3 + 6x^2 + 30x - 16
\]

### Step 2: Second Division

Next, divide the first term of the new expression by the first term of the denominator:

\[
\frac{4x^4}{x^3} = 4x
\]

**Quotient so far: \(x^2 + 4x\)**

Multiply \(4x\) by the entire denominator:

\[
4x \times (x^3 - x^2 + x + 8) = 4x^4 - 4x^3 + 4x^2 + 32x
\]

Subtract this from the current expression:

\[
(4x^4 - 6x^3 + 6x^2 + 30x - 16) - (4x^4 - 4x^3 + 4x^2 + 32x)
\]

This gives:

\[
(-6x^3 + 4x^3) + (6x^2 - 4x^2) + (30x - 32x) - 16
\]

Simplifying:

\[
-2x^3 + 2x^2 - 2x - 16
\]

### Step 3: Third Division

Divide the first term of this new expression by the first term of the denominator:

\[
\frac{-2x^3}{x^3} = -2
\]

**Quotient so far: \(x^2 + 4x - 2\)**

Multiply \(-2\) by the entire denominator:

\[
-2 \times (x^3 - x^2 + x + 8) = -2x^3 + 2x^2 - 2x - 16
\]

Subtract this:

\[
(-2x^3 + 2x^2 - 2x - 16) - (-2x^3 + 2x^2 - 2x - 16)
\]

The result is:

\[
0
\]

### Final Answer

Since the remainder is 0, the quotient is:

\[
\boxed{x^2 + 4x - 2}
\]

There is no remainder.

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**Related Questions:**

1. How do you check if your polynomial division answer is correct?
2. What happens when the remainder is not zero?
3. Can you always divide polynomials like this?
4. How does the degree of the polynomial affect the

Learn how to solve the polynomial division problem to find both the quotient and remainder using detailed step-by-step solutions. This problem involves dividing a polynomial by another polynomial, demonstrating the application of quotient and remainder theorem. Suitable for high school students.

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