To divide \( 2x^3 - 5x^2 - 8x + 15 \) by \( x + 5 \), we'll use polynomial long division.

### Step 1: Divide the leading term
The first term of the dividend is \( 2x^3 \), and the leading term of the divisor is \( x \). Divide \( 2x^3 \) by \( x \):

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

### Step 2: Multiply and subtract
Now multiply \( 2x^2 \) by \( x + 5 \):

\[
2x^2(x + 5) = 2x^3 + 10x^2
\]

Subtract \( 2x^3 + 10x^2 \) from the original polynomial:

\[
(2x^3 - 5x^2 - 8x + 15) - (2x^3 + 10x^2) = -15x^2 - 8x + 15
\]

### Step 3: Divide the next term
Now, divide \( -15x^2 \) by \( x \):

\[
\frac{-15x^2}{x} = -15x
\]

### Step 4: Multiply and subtract
Multiply \( -15x \) by \( x + 5 \):

\[
-15x(x + 5) = -15x^2 - 75x
\]

Subtract \( -15x^2 - 75x \) from \( -15x^2 - 8x + 15 \):

\[
(-15x^2 - 8x + 15) - (-15x^2 - 75x) = 67x + 15
\]

### Step 5: Divide the next term
Now divide \( 67x \) by \( x \):

\[
\frac{67x}{x} = 67
\]

### Step 6: Multiply and subtract
Multiply \( 67 \) by \( x + 5 \):

\[
67(x + 5) = 67x + 335
\]

Subtract \( 67x + 335 \) from \( 67x + 15 \):

\[
(67x + 15) - (67x + 335) = -320
\]

### Final Answer
The quotient is \( 2x^2 - 15x + 67 \) and the remainder is \( -320 \).

Thus, the division of \( 2x^3 - 5x^2 - 8x + 15 \) by \( x + 5 \) is:

\[
2x^2 - 15x + 67 - \frac{320}{x + 5}
\]

Would you like more details on any step?

### Related Questions
1. How do you perform synthetic division for this problem?
2. Can you verify the result by multiplying back?
3. What happens if the remainder is zero in polynomial division?
4. How is polynomial division related to factoring?
5. What is the difference between polynomial long division and synthetic division?

### Tip
Always align terms by degree when performing polynomial long division for better clarity and accuracy.

Master the process of polynomial long division by learning how to divide 2x³ - 5x² - 8x + 15 by x + 5. This comprehensive guide covers the algebraic concepts and steps involved, suitable for students in Grades 10-12.

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