You seem to be asking to divide the polynomial expression \( 2x^3 + 7x^2 - 5x + 2 \) by \( x - 2 \). This can be done using polynomial long division or synthetic division. I'll proceed with the polynomial long division method to solve this.

### Polynomial Long Division

1. **Divide the leading term of the dividend by the leading term of the divisor**:
   \[
   \frac{2x^3}{x} = 2x^2
   \]
   Multiply \( 2x^2 \) by \( x - 2 \), and subtract the result from the original polynomial:
   \[
   2x^3 - 4x^2
   \]
   Subtracting:
   \[
   (2x^3 + 7x^2) - (2x^3 - 4x^2) = 11x^2
   \]

2. **Repeat the process**:
   Divide \( 11x^2 \) by \( x \):
   \[
   \frac{11x^2}{x} = 11x
   \]
   Multiply \( 11x \) by \( x - 2 \), and subtract:
   \[
   11x^2 - 22x
   \]
   Subtracting:
   \[
   (11x^2 - 5x) - (11x^2 - 22x) = 17x
   \]

3. **Continue**:
   Divide \( 17x \) by \( x \):
   \[
   \frac{17x}{x} = 17
   \]
   Multiply \( 17 \) by \( x - 2 \), and subtract:
   \[
   17x - 34
   \]
   Subtracting:
   \[
   (17x + 2) - (17x - 34) = 36
   \]

So, the quotient is:
\[
2x^2 + 11x + 17
\]
And the remainder is:
\[
36
\]

Thus, the result of the division is:
\[
2x^2 + 11x + 17 + \frac{36}{x - 2}
\]

Would you like more details or have any questions?

### Related Questions:
1. How does synthetic division differ from long division?
2. What are the practical applications of dividing polynomials?
3. Can polynomial division always be done this way, regardless of the degree of the polynomial?
4. How would you apply the Remainder Theorem in this case?
5. What would happen if the divisor was a quadratic polynomial instead of linear?

**Tip:** Always double-check your subtraction steps during polynomial long division to avoid simple mistakes.

Learn how to solve the polynomial division problem: 2x^3 + 7x^2 - 5x + 2 divided by x - 2, using polynomial long division. This step-by-step solution covers key algebraic concepts and includes the quotient and remainder. Suitable for students in Grades 9-12.

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