To divide the polynomial \((4x^5 - 6x^4 + 5x^2 - x - 10)\) by \((2x^2 + 3)\), we'll use polynomial long division. Here are the steps:

### Step 1: Set up the division
We write the division as:

\[
\frac{4x^5 - 6x^4 + 5x^2 - x - 10}{2x^2 + 3}
\]

### Step 2: Divide the first term of the dividend by the first term of the divisor
Divide \(4x^5\) by \(2x^2\):

\[
\frac{4x^5}{2x^2} = 2x^3
\]

This will be the first term of the quotient.

### Step 3: Multiply the entire divisor by this term and subtract
Multiply \(2x^3\) by the divisor \(2x^2 + 3\):

\[
(2x^3)(2x^2 + 3) = 4x^5 + 6x^3
\]

Now subtract this from the original polynomial:

\[
(4x^5 - 6x^4 + 5x^2 - x - 10) - (4x^5 + 6x^3) = -6x^4 - 6x^3 + 5x^2 - x - 10
\]

### Step 4: Repeat the process
Now, divide the new leading term \(-6x^4\) by \(2x^2\):

\[
\frac{-6x^4}{2x^2} = -3x^2
\]

Multiply \(-3x^2\) by the divisor \(2x^2 + 3\):

\[
(-3x^2)(2x^2 + 3) = -6x^4 - 9x^2
\]

Subtract this from the current polynomial:

\[
(-6x^4 - 6x^3 + 5x^2 - x - 10) - (-6x^4 - 9x^2) = -6x^3 + 14x^2 - x - 10
\]

### Step 5: Continue dividing
Next, divide \(-6x^3\) by \(2x^2\):

\[
\frac{-6x^3}{2x^2} = -3x
\]

Multiply \(-3x\) by the divisor \(2x^2 + 3\):

\[
(-3x)(2x^2 + 3) = -6x^3 - 9x
\]

Subtract this from the polynomial:

\[
(-6x^3 + 14x^2 - x - 10) - (-6x^3 - 9x) = 14x^2 + 8x - 10
\]

### Step 6: Divide again
Divide \(14x^2\) by \(2x^2\):

\[
\frac{14x^2}{2x^2} = 7
\]

Multiply \(7\) by the divisor \(2x^2 + 3\):

\[
7(2x^2 + 3) = 14x^2 + 21
\]

Subtract this from the polynomial:

\[
(14x^2 + 8x - 10) - (14x^2 + 21) = 8x - 31
\]

### Final Answer:
The quotient is:

\[
2x^3 - 3x^2 - 3x + 7
\]

And the remainder is:

\[
\frac{8x - 31}{2x^2 + 3}
\]

So, the complete expression is:

\[
\frac{4x^5 - 6x^4 + 5x^2 - x - 10}{2x^2 + 3} = 2x^3 - 3x^2 - 3x + 7 + \frac{8x - 31}{2x^2 + 3}
\]

Would you like any additional details or explanations?

### Related Questions:
1. How do you perform synthetic division on polynomials?
2. What are the applications of polynomial division in calculus?
3. How can you determine if a polynomial is divisible by another polynomial without a remainder?
4. What is the significance of the remainder in polynomial division?
5. Can polynomial division be used to find roots of polynomials?

### Tip:
Always double-check each step in polynomial division to avoid mistakes, especially when subtracting polynomials during the process.

Learn how to divide the polynomial (4x^5 - 6x^4 + 5x^2 - x - 10) by (2x^2 + 3) using polynomial long division. Follow a step-by-step solution and understand the process of dividing polynomials. Suitable for high school students.

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