Let's perform the long division of \((2x^4 - 19x^2 + x^3 + 18x + 5) \div (2x + 5)\).

### Step 1: Arrange the polynomial in descending order of degree
First, we rearrange the polynomial in descending order of the degrees of \(x\):
\[
2x^4 + x^3 - 19x^2 + 18x + 5
\]

### Step 2: Divide the leading terms
- Divide the leading term of the dividend \(2x^4\) by the leading term of the divisor \(2x\):
  \[
  \frac{2x^4}{2x} = x^3
  \]
- So, the first term of the quotient is \(x^3\).

### Step 3: Multiply and subtract
- Multiply \(x^3\) by the divisor \(2x + 5\):
  \[
  x^3 \cdot (2x + 5) = 2x^4 + 5x^3
  \]
- Subtract this from the original polynomial:
  \[
  (2x^4 + x^3 - 19x^2 + 18x + 5) - (2x^4 + 5x^3) = -4x^3 - 19x^2 + 18x + 5
  \]

### Step 4: Repeat the process
- Divide the new leading term \(-4x^3\) by \(2x\):
  \[
  \frac{-4x^3}{2x} = -2x^2
  \]
- The next term of the quotient is \(-2x^2\).

### Step 5: Multiply and subtract
- Multiply \(-2x^2\) by the divisor \(2x + 5\):
  \[
  -2x^2 \cdot (2x + 5) = -4x^3 - 10x^2
  \]
- Subtract this from the current polynomial:
  \[
  (-4x^3 - 19x^2 + 18x + 5) - (-4x^3 - 10x^2) = -9x^2 + 18x + 5
  \]

### Step 6: Repeat the process again
- Divide the new leading term \(-9x^2\) by \(2x\):
  \[
  \frac{-9x^2}{2x} = -\frac{9}{2}x
  \]
- The next term of the quotient is \(-\frac{9}{2}x\).

### Step 7: Multiply and subtract
- Multiply \(-\frac{9}{2}x\) by the divisor \(2x + 5\):
  \[
  -\frac{9}{2}x \cdot (2x + 5) = -9x^2 - \frac{45}{2}x
  \]
- Subtract this from the current polynomial:
  \[
  \left(-9x^2 + 18x + 5\right) - \left(-9x^2 - \frac{45}{2}x\right) = \frac{81}{2}x + 5
  \]

### Step 8: Final division step
- Divide the new leading term \(\frac{81}{2}x\) by \(2x\):
  \[
  \frac{\frac{81}{2}x}{2x} = \frac{81}{4}
  \]
- The next term of the quotient is \(\frac{81}{4}\).

### Step 9: Multiply and subtract
- Multiply \(\frac{81}{4}\) by the divisor \(2x + 5\):
  \[
  \frac{81}{4} \cdot (2x + 5) = \frac{81}{2}x + \frac{405}{4}
  \]
- Subtract this from the current polynomial:
  \[
  \left(\frac{81}{2}x + 5\right) - \left(\frac{81}{2}x + \frac{405}{4}\right) = 5 - \frac{405}{4} = \frac{-385}{4}
  \]

### Final Answer
The quotient is:
\[
\boxed{x^3 - 2x^2 - \frac{9}{2}x + \frac{81}{4}}
\]
And the remainder is:
\[
\boxed{\frac{-385}{4}}
\]

So, the final result of the division is:
\[
x^3 - 2x^2 - \frac{9}{2}x + \frac{81}{4} + \frac{-385}{4(2x + 5)}
\]

Would you like further clarification on any step, or do you have any other questions?

Here are 5 related questions you might find interesting:
1. How do you handle polynomial division when the divisor has a degree greater than one?
2. Can you explain how to perform synthetic division with a linear binomial divisor?
3. What is the Remainder Theorem, and how can it be applied to polynomial division?
4. How can you check if a polynomial is divisible by another polynomial without remainder?
5. How do you perform polynomial division when the divisor is not in the simplest form?

**Tip:** Always double-check your polynomial terms and coefficients after subtraction during long division to avoid errors in the final quotient.

Explore a detailed step-by-step solution for polynomial division of (2x^4 - 19x^2 + x^3 + 18x + 5) ÷ (2x + 5). This division involves arranging the polynomial in descending order, dividing leading terms, multiplying and subtracting, and identifying the quotient and remainder. Suitable for advanced high school students and college-level learners.

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