solve with formula and solution (2x³+ 17x²+38x+15) ÷ (5+2x) long division

To solve \((2x³ + 17x² + 38x + 15) \div (2x + 5)\) using long division, follow these steps carefully:

### Step-by-Step Long Division Process:

**Dividend**: \(2x^3 + 17x^2 + 38x + 15\)  
**Divisor**: \(2x + 5\)

#### Step 1: Divide the first term of the dividend by the first term of the divisor
\[
\frac{2x^3}{2x} = x^2
\]
Now, multiply \(x^2\) by the divisor \(2x + 5\):
\[
x^2 \cdot (2x + 5) = 2x^3 + 5x^2
\]

#### Step 2: Subtract this from the original polynomial
\[
(2x^3 + 17x^2 + 38x + 15) - (2x^3 + 5x^2) = 12x^2 + 38x + 15
\]

#### Step 3: Divide the first term of the new polynomial by the first term of the divisor
\[
\frac{12x^2}{2x} = 6x
\]
Now, multiply \(6x\) by the divisor \(2x + 5\):
\[
6x \cdot (2x + 5) = 12x^2 + 30x
\]

#### Step 4: Subtract this from the new polynomial
\[
(12x^2 + 38x + 15) - (12x^2 + 30x) = 8x + 15
\]

#### Step 5: Divide the first term of the new polynomial by the first term of the divisor
\[
\frac{8x}{2x} = 4
\]
Now, multiply \(4\) by the divisor \(2x + 5\):
\[
4 \cdot (2x + 5) = 8x + 20
\]

#### Step 6: Subtract this from the new polynomial
\[
(8x + 15) - (8x + 20) = -5
\]

### Final Answer:
The quotient is:
\[
x^2 + 6x + 4
\]
And the remainder is:
\[
-5
\]
So, the final result is:
\[
x^2 + 6x + 4 - \frac{5}{2x + 5}
\]

Would you like more details or further clarifications on any step?

### Related Questions:
1. How is polynomial long division different from synthetic division?
2. Can you explain the concept of remainders in polynomial division?
3. What if the divisor was of higher degree than the dividend? How would the result change?
4. How can this result be verified by multiplying back the quotient and divisor?
5. What are some applications of polynomial division in real-world problems?

### Tip:
Always double-check your subtraction and multiplication at each step to avoid small errors in long division!

Learn how to solve the polynomial division of (2x³ + 17x² + 38x + 15) by (2x + 5) using the long division method. This step-by-step solution covers key algebraic concepts, including polynomial division and the polynomial division algorithm, suitable for high school students in Grades 9-12.

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