(×³+4×²-×-25)÷(×+5) synthetic division 

To solve \((x^3 + 4x^2 - x - 25) \div (x + 5)\) using synthetic division, we follow these steps:

### Step 1: Set up the synthetic division
- We divide \(x^3 + 4x^2 - x - 25\) by \(x + 5\), so the divisor's root is \(-5\).
- The coefficients of the polynomial are \(1\) (for \(x^3\)), \(4\) (for \(x^2\)), \(-1\) (for \(x\)), and \(-25\) (constant term).

Thus, the setup looks like this:

\[
\begin{array}{r|rrrr}
-5 & 1 & 4 & -1 & -25 \\
   &   &   &   &    \\
\end{array}
\]

### Step 2: Perform the synthetic division

1. Bring down the first coefficient \(1\).
   
   \[
   \begin{array}{r|rrrr}
   -5 & 1 & 4 & -1 & -25 \\
      &   &   &   &    \\
      & 1 &   &   &    \\
   \end{array}
   \]

2. Multiply \(-5\) by \(1\) (which gives \(-5\)) and add it to \(4\), getting \((4 - 5 = -1)\).

   \[
   \begin{array}{r|rrrr}
   -5 & 1 & 4 & -1 & -25 \\
      &   & -5 &   &    \\
      & 1 & -1 &   &    \\
   \end{array}
   \]

3. Multiply \(-5\) by \(-1\) (which gives \(5\)) and add it to \(-1\), getting \((-1 + 5 = 4)\).

   \[
   \begin{array}{r|rrrr}
   -5 & 1 & 4 & -1 & -25 \\
      &   & -5 & 5 &    \\
      & 1 & -1 & 4 &    \\
   \end{array}
   \]

4. Multiply \(-5\) by \(4\) (which gives \(-20\)) and add it to \(-25\), getting \((-25 - 20 = -45)\).

   \[
   \begin{array}{r|rrrr}
   -5 & 1 & 4 & -1 & -25 \\
      &   & -5 & 5 & -20 \\
      & 1 & -1 & 4 & -45 \\
   \end{array}
   \]

### Step 3: Write the quotient and remainder
- The quotient is \(x^2 - x + 4\).
- The remainder is \(-45\).

Thus, the result of the division is:
\[
\frac{x^3 + 4x^2 - x - 25}{x + 5} = x^2 - x + 4 - \frac{45}{x+5}
\]

Would you like more details or further clarification?

### Related Questions:
1. How do you perform long division on polynomials?
2. What is the relationship between synthetic and long division?
3. How can synthetic division be used to factor polynomials?
4. What happens when dividing polynomials with complex roots?
5. How is synthetic division applied in finding polynomial roots?

### Tip:
When using synthetic division, always check that the divisor is linear (\(x + a\)) for the method to work effectively.

(x³ + 4x² - x - 25) ÷ (x + 5) synthetic division

Learn how to use synthetic division to divide the polynomial (x³ + 4x² - x - 25) by (x + 5) with a step-by-step breakdown. This solution covers key algebraic concepts in polynomial division and synthetic division, perfect for students in Grades 10-12.

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