Sure, let's complete the factoring process for the polynomial [tex]\( A^4 + 2A^3B - 3A - 6B \)[/tex] step-by-step:
1. Identify possible groupings: Start by examining the polynomial [tex]\( A^4 + 2A^3B - 3A - 6B \)[/tex] and look for possible groupings of terms that could facilitate factoring:
Group terms as:
[tex]\[
(A^4 + 2A^3B) + (-3A - 6B)
\][/tex]
2. Factor out common factors from each group:
- From the first group, [tex]\( A^4 + 2A^3B \)[/tex]:
We can factor out [tex]\( A^3 \)[/tex]:
[tex]\[
A^3(A + 2B)
\][/tex]
- From the second group, [tex]\( -3A - 6B \)[/tex]:
We can factor out [tex]\( -3 \)[/tex]:
[tex]\[
-3(A + 2B)
\][/tex]
3. Combine the factored groups:
Now we can see a common factor [tex]\( (A + 2B) \)[/tex] in both terms:
[tex]\[
A^3(A + 2B) - 3(A + 2B)
\][/tex]
4. Factor out the common binomial:
Since [tex]\( (A + 2B) \)[/tex] is common in both terms, we can factor it out:
[tex]\[
(A + 2B)(A^3 - 3)
\][/tex]
Therefore, the factored form of the polynomial [tex]\( A^4 + 2A^3B - 3A - 6B \)[/tex] is:
[tex]\[
(A + 2B)(A^3 - 3)
\][/tex]
