To divide \((2x^3 + 17x^2 + 38x + 15)\) by \((2x + 5)\), we can use polynomial long division. Let's go through it step by step:

### Step 1: Divide the leading terms
The leading term of the dividend is \(2x^3\), and the leading term of the divisor is \(2x\). Divide these:
\[
\frac{2x^3}{2x} = x^2
\]
This gives the first term of the quotient: \(x^2\).

### Step 2: Multiply and subtract
Now, multiply \(x^2\) by the divisor \((2x + 5)\):
\[
x^2(2x + 5) = 2x^3 + 5x^2
\]
Subtract this from the original polynomial:
\[
(2x^3 + 17x^2 + 38x + 15) - (2x^3 + 5x^2) = 12x^2 + 38x + 15
\]

### Step 3: Divide the new leading terms
The new leading term is \(12x^2\). Divide by the leading term of the divisor:
\[
\frac{12x^2}{2x} = 6x
\]
So, the next term in the quotient is \(6x\).

### Step 4: Multiply and subtract
Now, multiply \(6x\) by \((2x + 5)\):
\[
6x(2x + 5) = 12x^2 + 30x
\]
Subtract this from the current polynomial:
\[
(12x^2 + 38x + 15) - (12x^2 + 30x) = 8x + 15
\]

### Step 5: Divide the new leading terms
Now, divide \(8x\) by \(2x\):
\[
\frac{8x}{2x} = 4
\]
So, the next term in the quotient is \(4\).

### Step 6: Multiply and subtract
Now, multiply \(4\) by \((2x + 5)\):
\[
4(2x + 5) = 8x + 20
\]
Subtract this from the current polynomial:
\[
(8x + 15) - (8x + 20) = -5
\]

### Final result:
The quotient is \(x^2 + 6x + 4\), and the remainder is \(-5\). Thus, we can express the result of the division as:
\[
\frac{2x^3 + 17x^2 + 38x + 15}{2x + 5} = x^2 + 6x + 4 - \frac{5}{2x + 5}
\]

Would you like more details or have any questions about the process?

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Here are 5 related questions you might find interesting:
1. How do you divide polynomials using synthetic division?
2. Can you factor the quadratic \(x^2 + 6x + 4\)?
3. What are the roots of the polynomial \(2x^3 + 17x^2 + 38x + 15\)?
4. How do you find the remainder when dividing two polynomials?
5. Can you apply the Remainder Theorem to check the remainder?

**Tip**: Always check the degree of the remainder. It should be less than the degree of the divisor in polynomial division.

Learn how to divide the polynomial (2x³+17x²+38x+15) by (2x+5) using polynomial long division. This step-by-step solution explains how to find the quotient and remainder, covering key algebraic concepts. Suitable for students in Grades 9-12.

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