Let's go through the process of factoring the polynomial [tex]\(-8x^3 - 2x^2 - 12x - 3\)[/tex] by grouping.
### Step 1: Group the Polynomial into Two Pairs
We can start by grouping the terms into two pairs:
[tex]\[ -8x^3 - 2x^2 \quad \text{and} \quad -12x - 3 \][/tex]
### Step 2: Factor Each Group
Now we factor out common factors from each group.
- For the first group [tex]\( -8x^3 - 2x^2 \)[/tex]:
[tex]\[ -8x^3 - 2x^2 = -2x^2(4x + 1) \][/tex]
- For the second group [tex]\( -12x - 3 \)[/tex]:
[tex]\[ -12x - 3 = -3(4x + 1) \][/tex]
### Step 3: Combine the Factors
Notice that both groups contain the common binomial factor [tex]\((4x + 1)\)[/tex]. Now we can factor out [tex]\((4x + 1)\)[/tex] from both groups:
We can express the original polynomial as:
[tex]\[ -8x^3 - 2x^2 - 12x - 3 = (-2x^2(4x + 1) - 3(4x + 1)) \][/tex]
Factor out [tex]\((4x + 1)\)[/tex] from both terms:
[tex]\[ = (4x + 1)(-2x^2 - 3) \][/tex]
### Conclusion
The factored form of the polynomial [tex]\(-8x^3 - 2x^2 - 12x - 3\)[/tex] by grouping is:
[tex]\[ \boxed{(4x + 1)(-2x^2 - 3)} \][/tex]
Thus, the correct answer is:
[tex]\[ \left(-2 x^2 - 3\right)(4 x + 1) \][/tex]
