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The problem in the image asks to determine the quotient \( q(x) \) and the remainder \( r(x) \) when dividing \( f(x) = x^3 + 2x^2 + 3x - 2 \) by \( g(x) = x^2 + 3x + 5 \), using polynomial long division.

### Step-by-Step Polynomial Long Division:

We are dividing \( f(x) = x^3 + 2x^2 + 3x - 2 \) by \( g(x) = x^2 + 3x + 5 \). The goal is to express \( f(x) \) as:
\[
f(x) = q(x) \cdot g(x) + r(x)
\]
where \( q(x) \) is the quotient and \( r(x) \) is the remainder, and the degree of \( r(x) \) should be less than that of \( g(x) \).

1. **First Division**:
   - Divide the leading term of \( f(x) \) by the leading term of \( g(x) \):
   \[
   \frac{x^3}{x^2} = x
   \]
   - Multiply \( x \) by \( g(x) \):
   \[
   x(x^2 + 3x + 5) = x^3 + 3x^2 + 5x
   \]
   - Subtract this from \( f(x) \):
   \[
   (x^3 + 2x^2 + 3x - 2) - (x^3 + 3x^2 + 5x) = -x^2 - 2x - 2
   \]

2. **Second Division**:
   - Divide the new leading term by the leading term of \( g(x) \):
   \[
   \frac{-x^2}{x^2} = -1
   \]
   - Multiply \( -1 \) by \( g(x) \):
   \[
   -1(x^2 + 3x + 5) = -x^2 - 3x - 5
   \]
   - Subtract this from the previous result:
   \[
   (-x^2 - 2x - 2) - (-x^2 - 3x - 5) = x + 3
   \]

### Final Expression:
Now, we have:
\[
f(x) = (x - 1) \cdot g(x) + (x + 3)
\]
Thus, the quotient is \( q(x) = x - 1 \) and the remainder is \( r(x) = x + 3 \).

### Final Answer:
\[
[q(x), r(x)] = [x - 1, x + 3]
\]

Would you like further details or clarifications? Here are 5 related questions for practice:

1. What is the remainder if we divide \( x^3 + 5x^2 + 2x + 1 \) by \( x^2 + 1 \)?
2. Can you perform synthetic division for dividing a cubic polynomial by a linear factor?
3. How does the degree of the remainder compare to the degree of the divisor in polynomial division?
4. What are the conditions for a polynomial to be divisible by another polynomial without remainder?
5. How is polynomial division related to the factor theorem?

**Tip**: Always ensure the degree of the remainder is less than the degree of the divisor during polynomial division.

Determine the quotient q(x) and the remainder r(x) when dividing f(x) = x^3 + 2x^2 + 3x - 2 by g(x) = x^2 + 3x + 5.

Learn how to solve a polynomial division problem by dividing f(x) = x^3 + 2x^2 + 3x - 2 by g(x) = x^2 + 3x + 5. Understand how to find the quotient and remainder using the polynomial division algorithm. Suitable for students in Grades 10-12.

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