Use the long division method to find the result when 8, x, cubed, plus, 30, x, squared, plus, 23, x, plus, 48x 
3
 +30x 
2
 +23x+4 is divided by 4, x, plus, 14x+1.

To perform long division of polynomials, we divide the given dividend \(8x^3 + 30x^2 + 23x + 4\) by the divisor \(4x + 1\). Let's go step by step:

### Step 1: Divide the leading terms
- Divide the leading term of the dividend by the leading term of the divisor.
  
\[
\frac{8x^3}{4x} = 2x^2
\]

### Step 2: Multiply the divisor by this quotient
- Multiply \(2x^2\) by the entire divisor \(4x + 1\):

\[
2x^2 \cdot (4x + 1) = 8x^3 + 2x^2
\]

### Step 3: Subtract the result from the original polynomial
- Subtract \((8x^3 + 2x^2)\) from the original dividend:

\[
(8x^3 + 30x^2 + 23x + 4) - (8x^3 + 2x^2) = 28x^2 + 23x + 4
\]

### Step 4: Divide the leading terms again
- Now, divide the leading term of the new polynomial \(28x^2\) by \(4x\):

\[
\frac{28x^2}{4x} = 7x
\]

### Step 5: Multiply the divisor by this quotient
- Multiply \(7x\) by the entire divisor \(4x + 1\):

\[
7x \cdot (4x + 1) = 28x^2 + 7x
\]

### Step 6: Subtract again
- Subtract \((28x^2 + 7x)\) from \((28x^2 + 23x + 4)\):

\[
(28x^2 + 23x + 4) - (28x^2 + 7x) = 16x + 4
\]

### Step 7: Divide the leading terms again
- Now, divide \(16x\) by \(4x\):

\[
\frac{16x}{4x} = 4
\]

### Step 8: Multiply the divisor by this quotient
- Multiply \(4\) by the divisor \(4x + 1\):

\[
4 \cdot (4x + 1) = 16x + 4
\]

### Step 9: Subtract again
- Subtract \((16x + 4)\) from \((16x + 4)\):

\[
(16x + 4) - (16x + 4) = 0
\]

### Final Result
There is no remainder, and the quotient is:

\[
2x^2 + 7x + 4
\]

---

Would you like to go over any details, or do you have any questions?

Here are 5 related questions to expand your understanding:

1. How does synthetic division compare to long division for polynomial division?
2. What happens if the remainder is non-zero when dividing polynomials?
3. Can polynomial division be used to factor polynomials? If so, how?
4. How would you perform long division for higher-degree polynomials (like quartic polynomials)?
5. How is the division of polynomials related to finding asymptotes in rational functions?

**Tip**: Always ensure the divisor's leading term is factored out correctly before proceeding with division to avoid errors.

Use the long division method to find the result when 8x^3 + 30x^2 + 23x + 4 is divided by 4x + 1.

Learn how to perform polynomial long division with a step-by-step solution for dividing 8x^3 + 30x^2 + 23x + 4 by 4x + 1. This problem covers key algebraic concepts, including division of polynomials and applying the long division method. Suitable for students in Grades 9-11.

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