The expression you've provided is:

\[
\frac{3x^3 + 14x^2 - 19x + 25}{x + 6}
\]

To simplify or divide this polynomial by \(x + 6\), we can use **polynomial division** (similar to long division) or **synthetic division**. Let's use synthetic division, which is generally quicker for this type of problem.

### Step-by-Step Solution:

1. **Identify the coefficients** from the numerator polynomial \(3x^3 + 14x^2 - 19x + 25\). These are \(3\), \(14\), \(-19\), and \(25\).

2. **Use the root** of \(x + 6 = 0\), which is \(x = -6\), for synthetic division.

3. **Perform synthetic division**:

   - Write down the coefficients: \(3, 14, -19, 25\).
   - Bring down the first coefficient \(3\) as it is.
   - Multiply \(3\) by \(-6\) (the root) and add it to the next coefficient \(14\).
   - Continue this process for all coefficients.

\[
\begin{array}{r|rrrr}
-6 & 3 & 14 & -19 & 25 \\
   &   & -18 & 24 & -30 \\
\hline
   & 3 & -4 & 5 & -5 \\
\end{array}
\]

The result of the synthetic division is \(3x^2 - 4x + 5\) with a remainder of \(-5\).

### Final Answer:
The division yields:

\[
\frac{3x^3 + 14x^2 - 19x + 25}{x + 6} = 3x^2 - 4x + 5 - \frac{5}{x + 6}
\]

This result means the quotient is \(3x^2 - 4x + 5\) and the remainder is \(-5\), so you have the above expression as your final simplified form.

Would you like more details, or have any questions?

### Related Questions:
1. How do you perform synthetic division for a cubic polynomial?
2. What is the relationship between the remainder and the divisor in polynomial division?
3. Can synthetic division be used for non-linear divisors?
4. What are the key differences between long division and synthetic division in polynomials?
5. How can the remainder theorem be applied to check polynomial division results?

### Tip:
When using synthetic division, remember that the divisor must be in the form \(x - c\) where \(c\) is a constant.

Master the process of dividing the polynomial 3x^3 + 14x^2 - 19x + 25 by x + 6 using synthetic division. This solution provides a step-by-step approach, involving key algebraic concepts, formulas, and the Remainder Theorem. Perfect for students in Grades 10-12.

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