https://file.aiquickdraw.com/mathbot/file/1726155945551okd2ufny.jpg

The problem shown in the image is a polynomial division problem using synthetic division. The given expression to divide is:

\[
(3x^2 + 8x + 5x^3 + 2x^4 + 12) \div (2x + 3)
\]

### Steps to solve using synthetic division:

1. **Rearrange the dividend** in standard form (highest to lowest degree). So, rewrite the polynomial \( 2x^4 + 5x^3 + 3x^2 + 8x + 12 \).

2. **Set up the synthetic division**:
    - You need to divide by \( 2x + 3 \). First, rewrite this as \( x - r \) form. In this case, \( 2x + 3 = 0 \) gives \( x = -\frac{3}{2} \).
    
3. **List coefficients** of the polynomial \( 2x^4 + 5x^3 + 3x^2 + 8x + 12 \), which are: \( [2, 5, 3, 8, 12] \).

4. **Synthetic division process**:
    - Perform synthetic division using \( r = -\frac{3}{2} \).

Let me perform the synthetic division process for you.The result of the synthetic division process gives the quotient and remainder:

\[
2x^3 + 2x^2 + 0x + 8 \quad \text{with a remainder of } 0
\]

Thus, the quotient of the division \( (2x^4 + 5x^3 + 3x^2 + 8x + 12) \div (2x + 3) \) is:

\[
2x^3 + 2x^2 + 8
\]

The division results in a zero remainder, meaning the division is exact.

### Do you have any questions about this? Would you like more details?

Here are five related questions:
1. What happens when there is a remainder in synthetic division?
2. How does synthetic division differ from long division of polynomials?
3. Can synthetic division be used when dividing by a non-linear divisor?
4. How does the remainder theorem relate to synthetic division?
5. What is the importance of arranging polynomials in descending order of powers?

**Tip:** Always make sure to factor the divisor completely to apply synthetic division correctly.

Divide the polynomials using synthetic division: (3x^2 + 8x + 5x^3 + 2x^4 + 12) ÷ (2x + 3).

Learn how to divide polynomials using synthetic division with a step-by-step solution to the problem (2x^4 + 5x^3 + 3x^2 + 8x + 12) ÷ (2x + 3). Explore core algebraic concepts, including synthetic division and the remainder theorem. Suitable for students in Grades 10-12.

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