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To solve these polynomial division problems using long division, I'll work through each one step by step. Let's start with the first problem:

### 1. \((5x^2 - 15 - 17x) \div (x - 4)\)

#### Step 1: Arrange the dividend and divisor in standard form.
Dividend: \(5x^2 - 17x - 15\)  
Divisor: \(x - 4\)

#### Step 2: Divide the first term of the dividend by the first term of the divisor.
\[
\frac{5x^2}{x} = 5x
\]
This is the first term of the quotient.

#### Step 3: Multiply the entire divisor by the first term of the quotient.
\[
5x \times (x - 4) = 5x^2 - 20x
\]

#### Step 4: Subtract the result from the original dividend.
\[
(5x^2 - 17x - 15) - (5x^2 - 20x) = 3x - 15
\]

#### Step 5: Repeat the process with the new dividend.
Divide the first term of the new dividend by the first term of the divisor.
\[
\frac{3x}{x} = 3
\]

#### Step 6: Multiply the entire divisor by the new term of the quotient.
\[
3 \times (x - 4) = 3x - 12
\]

#### Step 7: Subtract again.
\[
(3x - 15) - (3x - 12) = -3
\]

### Final Answer for Problem 1:
\[
\text{Quotient: } 5x + 3
\text{ with a remainder of } -3
\]
Thus, \((5x^2 - 15 - 17x) \div (x - 4) = 5x + 3 - \frac{3}{x-4}\).

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Would you like me to continue with the next problems, or do you have any specific questions about this one?

### Related Questions:
1. How would you solve \((6x^2 - 16x^2 + 17x - 6) \div (3x - 2)\) using long division?
2. Can you perform synthetic division for the given polynomial division problem?
3. How can we verify the result of a polynomial division?
4. What are some alternative methods to long division for solving polynomial division?
5. How would the remainder be interpreted in the context of polynomial division?

### Tip:
When performing polynomial long division, always align like terms vertically to avoid mistakes in subtraction.

Learn how to solve the polynomial division problem (5x^2 - 15 - 17x) ÷ (x - 4) using long division. This detailed step-by-step solution covers polynomial long division techniques, suitable for advanced high school students.

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