You want to divide the polynomial \(3x^3 + 5x^2 - 42x - 40\) by \(x + 4\). We can perform this division using synthetic or long division. I'll use synthetic division here since it's efficient for dividing polynomials by linear terms like \(x + 4\).

1. **Step 1:** Identify the root of \(x + 4 = 0\), which is \(x = -4\).
2. **Step 2:** Set up synthetic division. The coefficients of the polynomial \(3x^3 + 5x^2 - 42x - 40\) are: \(3, 5, -42, -40\).

3. **Step 3:** Perform synthetic division using \(x = -4\).

\[
\begin{array}{r|rrrr}
-4 & 3 & 5 & -42 & -40 \\
   &   & -12 & 28 & 56  \\
\hline
   & 3 & -7  & -14  & 16  \\
\end{array}
\]

- First, bring down the \(3\).
- Multiply \(3\) by \(-4\) to get \(-12\), and add to \(5\) to get \(-7\).
- Multiply \(-7\) by \(-4\) to get \(28\), and add to \(-42\) to get \(-14\).
- Multiply \(-14\) by \(-4\) to get \(56\), and add to \(-40\) to get \(16\).

4. **Step 4:** The quotient is \(3x^2 - 7x - 14\), and the remainder is \(16\).

Thus, the division of \(3x^3 + 5x^2 - 42x - 40\) by \(x + 4\) gives:

\[
\frac{3x^3 + 5x^2 - 42x - 40}{x + 4} = 3x^2 - 7x - 14 + \frac{16}{x + 4}
\]

Would you like further explanation or have any questions?

Here are some related questions for you:
1. How would the division change if we divided by \(x - 4\) instead?
2. What is synthetic division, and how does it work in detail?
3. Can you solve for the roots of the quotient polynomial \(3x^2 - 7x - 14 = 0\)?
4. What happens if the remainder is zero in a polynomial division?
5. How does synthetic division compare to long division for polynomials?

**Tip:** Synthetic division works best for divisors of the form \(x - c\) where \(c\) is a constant.

Learn how to divide the polynomial 3x^3 + 5x^2 - 42x - 40 by x+4 using synthetic division. This problem explains the synthetic division process step-by-step, including how to calculate the quotient and remainder. Suitable for students in Grades 9-11.

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