Question: Use the long division method to find the result when $8x^3 + 18x^2 + 19x + 25$ is divided by $2x + 3$. If there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.
1. **State the problem:** Divide the polynomial $$8x^3 + 18x^2 + 19x + 25$$ by the binomial $$2x + 3$$ using long division.
2. **Recall the formula:** When dividing polynomials, the result can be expressed as $$\text{Dividend} = (\text{Divisor}) \times (\text{Quotient}) + \text{Remainder}$$.
3. **Step 1:** Divide the leading term of the dividend $$8x^3$$ by the leading term of the divisor $$2x$$:
$$\frac{8x^3}{2x} = 4x^2$$.
4. **Step 2:** Multiply the entire divisor by $$4x^2$$:
$$4x^2 \times (2x + 3) = 8x^3 + 12x^2$$.
5. **Step 3:** Subtract this from the dividend:
$$\begin{aligned}
&(8x^3 + 18x^2 + 19x + 25) - (8x^3 + 12x^2) \\
&= (8x^3 - 8x^3) + (18x^2 - 12x^2) + 19x + 25 \\
&= 6x^2 + 19x + 25
\end{aligned}$$
6. **Step 4:** Divide the leading term of the new polynomial $$6x^2$$ by $$2x$$:
$$\frac{6x^2}{2x} = 3x$$.
7. **Step 5:** Multiply the divisor by $$3x$$:
$$3x \times (2x + 3) = 6x^2 + 9x$$.
8. **Step 6:** Subtract this from the current polynomial:
$$\begin{aligned}
&(6x^2 + 19x + 25) - (6x^2 + 9x) \\
&= (6x^2 - 6x^2) + (19x - 9x) + 25 \\
&= 10x + 25
\end{aligned}$$
9. **Step 7:** Divide the leading term $$10x$$ by $$2x$$:
$$\frac{10x}{2x} = 5$$.
10. **Step 8:** Multiply the divisor by $$5$$:
$$5 \times (2x + 3) = 10x + 15$$.
11. **Step 9:** Subtract this from the current polynomial:
$$\begin{aligned}
&(10x + 25) - (10x + 15) \\
&= (10x - 10x) + (25 - 15) \\
&= 10
\end{aligned}$$
12. **Step 10:** Since the remainder $$10$$ is a constant and the divisor is degree 1, the division ends here.
13. **Final result:** The quotient is $$4x^2 + 3x + 5$$ and the remainder is $$10$$.
Express the result as:
$$\boxed{4x^2 + 3x + 5 + \frac{10}{2x + 3}}$$