To determine the eighth term of the given geometric sequence, we must first identify the general properties of the sequence and then use these properties to find the desired term.
### Step 1: Identify the Common Ratio
A geometric sequence is characterized by its constant common ratio [tex]\( r \)[/tex] between consecutive terms. Given the first three terms of the sequence:
1. [tex]\( a_1 = x + 3 \)[/tex]
2. [tex]\( a_2 = -2x^2 - 6x \)[/tex]
3. [tex]\( a_3 = 4x^3 + 12x^2 \)[/tex]
We can find the common ratio [tex]\( r \)[/tex] by dividing the second term by the first term:
[tex]\[ r = \frac{a_2}{a_1} = \frac{-2x^2 - 6x}{x + 3} \][/tex]
### Step 2: Verify the Common Ratio
Next, we verify the common ratio by ensuring that the ratio of the third term to the second term is the same:
[tex]\[ r = \frac{a_3}{a_2} = \frac{4x^3 + 12x^2}{-2x^2 - 6x} \][/tex]
Since the common ratio [tex]\( r \)[/tex] is verified to be consistent, we proceed to use it in calculating further terms.
### Step 3: General Term Formula
In a geometric sequence, the [tex]\( n \)[/tex]-th term [tex]\( a_n \)[/tex] can be expressed as:
[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]
For the eighth term [tex]\( a_8 \)[/tex]:
[tex]\[ a_8 = a_1 \cdot r^{7} \][/tex]
### Step 4: Calculate the Eighth Term
Using the identified common ratio [tex]\( r \)[/tex]:
[tex]\[ r = \frac{-2x^2 - 6x}{x + 3} \][/tex]
We substitute [tex]\( a_1 \)[/tex] and [tex]\( r \)[/tex] into the formula for [tex]\( a_8 \)[/tex]:
[tex]\[ a_8 = (x + 3) \left( \frac{-2x^2 - 6x}{x + 3} \right)^7 \][/tex]
### Step 5: Simplify the Expression
By simplifying the term, you find that:
[tex]\[ a_8 = -128 x^7 (x + 3) \][/tex]
### Conclusion
Thus, the eighth term of the given geometric sequence is:
[tex]\[ \boxed{-128 x^7 (x + 3)} \][/tex]
The eighth term matches one of the given options:
[tex]\[ \boxed{-128 x^8 - 384 x^7} \][/tex] which in another representation matches our simplified term [tex]\( -128 x^7 (x + 3) \)[/tex].
