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Let's solve the given problem step by step.

The polynomial is \( p(x) = 3x^2 - 6x - 5 \), and we are asked to divide \( p(x) \) by \( (x + 1) \). This gives us a quotient \( q(x) \).

### Step 1: Polynomial Division
We will perform the division \( p(x) \div (x + 1) \) using polynomial long division.

Given:
\[ p(x) = 3x^2 - 6x - 5 \]
and divisor \( d(x) = x + 1 \).

### Step 2: Polynomial Long Division
We divide \( 3x^2 - 6x - 5 \) by \( x + 1 \).

1. First, divide the leading term of the dividend \( 3x^2 \) by the leading term of the divisor \( x \), which gives \( 3x \).
2. Multiply \( 3x \) by \( x + 1 \), yielding \( 3x^2 + 3x \).
3. Subtract this from the original polynomial:
   \[
   (3x^2 - 6x - 5) - (3x^2 + 3x) = -9x - 5.
   \]
4. Now, divide \( -9x \) by \( x \), which gives \( -9 \).
5. Multiply \( -9 \) by \( x + 1 \), yielding \( -9x - 9 \).
6. Subtract this from \( -9x - 5 \):
   \[
   (-9x - 5) - (-9x - 9) = 4.
   \]

Thus, the quotient is \( q(x) = 3x - 9 \) and the remainder is 4.

### Step 3: Find \( q(6) \)
Now, substitute \( x = 6 \) into \( q(x) = 3x - 9 \):
\[
q(6) = 3(6) - 9 = 18 - 9 = 9.
\]

### Final Answer:
\( q(6) = 9 \).

Would you like more details on any step or further explanation? 

Here are 5 related questions to further explore:
1. How do you perform polynomial division with remainders?
2. What is the remainder theorem, and how does it apply here?
3. How do you factor polynomials like \( p(x) = 3x^2 - 6x - 5 \)?
4. Can we use synthetic division for the same problem, and how would it work?
5. How do polynomial division and the division algorithm relate?

**Tip:** Always check if the polynomial can be factored easily before proceeding with division; it can save time and simplify the process.

Considere o polinômio p(x) = 3x^2 - 6x - 5. Dividindo p(x) por (x + 1), obtém-se quociente q(x). Dessa forma, tem-se que q(6) é igual a ____.

Learn how to divide the polynomial p(x) = 3x^2 - 6x - 5 by (x + 1) to obtain q(x), and find the value of q(6). This problem involves key concepts of polynomial division and evaluation of functions. Suitable for students in Grades 9-11.

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