Certainly! Let's analyze and simplify the given expression step by step.
The given expression is:
[tex]\[ 8p^3 + q^3 + 12p^2q + 6pq^2 \][/tex]
We can approach this by trying to group terms in a way that reveals a common factor or a recognizable pattern. Notice that the terms involve powers and products of [tex]\(p\)[/tex] and [tex]\(q\)[/tex].
Let's rewrite the expression:
[tex]\[ 8p^3 + 12p^2q + 6pq^2 + q^3 \][/tex]
We can see that the terms can be grouped into two parts:
[tex]\[ (8p^3 + 12p^2q) + (6pq^2 + q^3) \][/tex]
First, focus on the group [tex]\(8p^3 + 12p^2q\)[/tex]:
- Both terms have a common factor of [tex]\(4p^2\)[/tex]:
[tex]\[ 8p^3 + 12p^2q = 4p^2(2p) + 4p^2(3q) = 4p^2(2p + 3q) \][/tex]
Next, look at the group [tex]\(6pq^2 + q^3\)[/tex]:
- Both terms have a common factor of [tex]\(q^2\)[/tex]:
[tex]\[ 6pq^2 + q^3 = q^2(6p) + q^2(q) = q^2(6p + q) \][/tex]
Now we have:
[tex]\[ 4p^2(2p + 3q) + q^2(6p + q) \][/tex]
Notice that the terms inside the parentheses are the same, suggesting a common binomial factor:
- Each grouped term has [tex]\(2p + 3q\)[/tex] as a factor:
[tex]\[ 4p^2(2p + 3q) + q^2(2p + 3q) = (4p^2 + q^2)(2p + 3q) \][/tex]
Thus, the original expression factors to:
[tex]\[ 4p^2(2p + 3q) + q^2(6p + q) \][/tex]
which gives us:
[tex]\[ (4p^2 + q^2)(2p + 3q) \][/tex]
Therefore, the simplified and factored expression is:
[tex]\[ 8p^3 + 12p^2q + 6pq^2 + q^3 = (4p^2 + q^2)(2p + 3q) \][/tex]
This detailed, step-by-step solution shows how to factorize the given polynomial expression.
