Given the polynomial: [tex]\(A^4+2 A^3 B-3 A-6 B\)[/tex]
Step 1: Factor by grouping.
Let's break the polynomial into two groups:
[tex]\[ A^4 + 2A^3B \][/tex] and [tex]\[-3A - 6B \][/tex]
Now, factor each group separately:
1. In the first group [tex]\( A^4 + 2A^3B \)[/tex]:
[tex]\[ A^4 + 2A^3B = A^3(A + 2B) \][/tex]
2. In the second group [tex]\(-3A - 6B\)[/tex]:
[tex]\[ -3A - 6B = -3(A + 2B) \][/tex]
Combining these factorizations gives:
[tex]\[ A^3(A + 2B) - 3(A + 2B) \][/tex]
Step 2: Factor out the common term [tex]\((A + 2B)\)[/tex]:
[tex]\[ (A + 2B)(A^3 - 3) \][/tex]
So, the filled-in blanks should be:
Step 1: [tex]\(A^{\mathbf{3}}(A+2 B)-3(\mathbf{A}+2 B)\)[/tex]
Step 2: [tex]\((A+2 \mathbf{B})\left(A^3-\mathbf{3}\right)\)[/tex]
The completed expression is:
[tex]\[ (A+2B)(A^3-3) \][/tex]
