https://file.aiquickdraw.com/mathbot/file/1727181369991vl84pj4q.jpg

The problem shown in the image asks you to divide the polynomial \(-3 - 14x + 10x^3\) by \(5x^2 - 4\), and to write the answer in the form:

\[
\text{Quotient} + \frac{\text{Remainder}}{5x^2 - 4}.
\]

Let's solve this using polynomial long division:

### Step 1: Set up the division
We are dividing:

\[
\frac{-3 - 14x + 10x^3}{5x^2 - 4}.
\]

First, we organize the terms of the polynomial in standard form: \(10x^3 - 14x - 3\).

### Step 2: Start the division
1. **Divide the leading term of the numerator by the leading term of the denominator:**
   \[
   \frac{10x^3}{5x^2} = 2x.
   \]

2. **Multiply \(2x\) by \(5x^2 - 4\):**
   \[
   2x(5x^2 - 4) = 10x^3 - 8x.
   \]

3. **Subtract this from the original polynomial:**
   \[
   (10x^3 - 14x - 3) - (10x^3 - 8x) = -6x - 3.
   \]

### Step 3: Continue the division
Now divide the new leading term \(-6x\) by \(5x^2\):

1. **Divide:**
   \[
   \frac{-6x}{5x^2} = -\frac{6}{5x}.
   \]

2. **Multiply \(-\frac{6}{5x}\) by \(5x^2 - 4\):**
   \[
   -\frac{6}{5x}(5x^2 - 4) = -6x + \frac{24}{5}.
   \]

3. **Subtract this from \(-6x - 3\):**
   \[
   (-6x - 3) - (-6x + \frac{24}{5}) = -3 - \frac{24}{5} = \frac{-15 - 24}{5} = \frac{-39}{5}.
   \]

### Step 4: Write the result
The quotient is \(2x - \frac{6}{5x}\), and the remainder is \(\frac{-39}{5}\).

So the final result is:

\[
2x - \frac{6}{5x} + \frac{-39}{5}{5x^2 - 4}.
\]

Do you need more details on any step? Let me know if you have any questions!

### Related Questions:
1. What is polynomial long division and how is it performed?
2. Can polynomial division always be done without a remainder?
3. How do you handle negative coefficients in polynomial division?
4. What are alternative methods for dividing polynomials?
5. What is synthetic division, and how does it differ from long division?

### Tip:
When dividing polynomials, always align the terms in decreasing order of the exponent, and subtract carefully to avoid sign errors.

Divide the polynomial (-3 - 14x + 10x^3) by (5x^2 - 4), and write the answer in the form: Quotient + (Remainder)/(5x^2 - 4).

Learn how to divide the polynomial (-3 - 14x + 10x^3) by (5x^2 - 4) using polynomial long division. This detailed solution involves dividing the leading terms and subtracting, resulting in a quotient of 2x - 6/(5x) and a remainder of -39/5. Suitable for students in Grades 10-12.

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