To divide the polynomial \( 5x - 6x^2 + 8x^3 + 46 \) by \( 2x + 3 \), we can use polynomial long division. Here’s the step-by-step process:

### Step 1: Arrange the terms
First, arrange both the dividend (the polynomial being divided) and the divisor in descending powers of \( x \):

\[
\text{Dividend: } 8x^3 - 6x^2 + 5x + 46
\]
\[
\text{Divisor: } 2x + 3
\]

### Step 2: Divide the first term of the dividend by the first term of the divisor
Divide \( 8x^3 \) by \( 2x \):

\[
\frac{8x^3}{2x} = 4x^2
\]

### Step 3: Multiply and subtract
Multiply the entire divisor \( 2x + 3 \) by \( 4x^2 \) and subtract the result from the dividend:

\[
(4x^2) \times (2x + 3) = 8x^3 + 12x^2
\]
\[
8x^3 - 6x^2 + 5x + 46 - (8x^3 + 12x^2) = -18x^2 + 5x + 46
\]

### Step 4: Repeat the process
Now, divide \( -18x^2 \) by \( 2x \):

\[
\frac{-18x^2}{2x} = -9x
\]

Multiply and subtract:

\[
(-9x) \times (2x + 3) = -18x^2 - 27x
\]
\[
-18x^2 + 5x + 46 - (-18x^2 - 27x) = 32x + 46
\]

### Step 5: Continue dividing
Now, divide \( 32x \) by \( 2x \):

\[
\frac{32x}{2x} = 16
\]

Multiply and subtract:

\[
(16) \times (2x + 3) = 32x + 48
\]
\[
32x + 46 - (32x + 48) = -2
\]

### Final Answer
The quotient is \( 4x^2 - 9x + 16 \) with a remainder of \( -2 \).

Thus, the final result is:

\[
\frac{8x^3 - 6x^2 + 5x + 46}{2x + 3} = 4x^2 - 9x + 16 - \frac{2}{2x + 3}
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you perform polynomial long division on different degrees of polynomials?
2. What are some common errors to avoid in polynomial long division?
3. How is synthetic division different from polynomial long division?
4. Can you solve a polynomial division where the remainder is a polynomial rather than a constant?
5. How would you graph the quotient \( 4x^2 - 9x + 16 \) from this division?

**Tip:** When performing polynomial long division, carefully align the terms by their degree to avoid mistakes in subtraction.

Learn how to perform polynomial long division on 8x^3 - 6x^2 + 5x + 46 divided by 2x + 3 with a detailed step-by-step solution. This problem covers polynomial division concepts and techniques, suitable for advanced high school students in Grades 11-12.

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