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To rewrite the polynomial \( p(x) = x^3 - 19x - 30 \) as a product of linear factors, given that \( x + 2 \) is a factor, we can use synthetic division or polynomial division to find the other factors.

Let's proceed with synthetic division:

1. **Set up the synthetic division** by dividing \( p(x) \) by \( x + 2 \). Since \( x + 2 = 0 \) when \( x = -2 \), we use \( -2 \) as the divisor.

2. The coefficients of \( p(x) = x^3 - 19x - 30 \) are \( 1 \) (for \( x^3 \)), \( 0 \) (for \( x^2 \)), \( -19 \) (for \( x \)), and \( -30 \) (constant term).

Let's perform synthetic division with \( -2 \):

\[
\begin{array}{r|rrrr}
-2 & 1 & 0 & -19 & -30 \\
   &   & -2 & 4 & 30 \\
\hline
   & 1 & -2 & -15 & 0 \\
\end{array}
\]

3. The remainder is \( 0 \), confirming \( x + 2 \) is a factor, and the quotient is \( x^2 - 2x - 15 \).

4. Now, factor \( x^2 - 2x - 15 \):
   \[
   x^2 - 2x - 15 = (x - 5)(x + 3)
   \]

5. Therefore, we can rewrite \( p(x) \) as a product of linear factors:
   \[
   p(x) = (x + 2)(x - 5)(x + 3)
   \]

### Final Answer
\[
p(x) = (x + 2)(x - 5)(x + 3)
\]

Would you like further explanation on any steps?

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### Related Questions

1. How do you confirm if a polynomial has a specific factor without synthetic division?
2. What if a polynomial has a repeated root? How would the factorization look?
3. Can synthetic division be used for polynomials with complex roots?
4. How do you factor higher-degree polynomials with four or more terms?
5. What is the remainder theorem and how does it relate to synthetic division?

### Tip
When using synthetic division, always double-check each step to avoid simple arithmetic errors, especially when there are zero coefficients.

Rewrite the polynomial \( p(x) = x^3 - 19x - 30 \) as a product of linear factors, given that \( x + 2 \) is a known factor.

Learn how to factor the polynomial p(x) = x^3 - 19x - 30 by rewriting it as a product of linear factors. This solution demonstrates the use of synthetic division and the Factor theorem, suitable for students in Grades 10-12.

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