Sure, let's find the completely factorized form of the expression [tex]\( 2a^4b^2c^3 + 8a^2b^3c^4 + 6a^4b^3c^3 \)[/tex].
First, identify common factors in all three terms. Notice that each term contains powers of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
### Step 1: Identify the Greatest Common Factor (GCF)
We need to find the GCF of the coefficients and variables in all the terms:
- For the coefficients [tex]\(2, 8, 6\)[/tex], the GCF is [tex]\(2\)[/tex].
- For [tex]\(a\)[/tex] terms: [tex]\(a^4\)[/tex], [tex]\(a^2\)[/tex], and [tex]\(a^4\)[/tex], the GCF is [tex]\(a^2\)[/tex].
- For [tex]\(b\)[/tex] terms: [tex]\(b^2\)[/tex], [tex]\(b^3\)[/tex], and [tex]\(b^3\)[/tex], the GCF is [tex]\(b^2\)[/tex].
- For [tex]\(c\)[/tex] terms: [tex]\(c^3\)[/tex], [tex]\(c^4\)[/tex], and [tex]\(c^3\)[/tex], the GCF is [tex]\(c^3\)[/tex].
Therefore, the GCF of the entire expression is [tex]\(2a^2b^2c^3\)[/tex].
### Step 2: Factor out the GCF
We factor [tex]\(2a^2b^2c^3\)[/tex] out of each term in the expression:
[tex]\[
2a^4b^2c^3 = 2a^2b^2c^3 \cdot a^2
\][/tex]
[tex]\[
8a^2b^3c^4 = 2a^2b^2c^3 \cdot 4bc
\][/tex]
[tex]\[
6a^4b^3c^3 = 2a^2b^2c^3 \cdot 3a^2b
\][/tex]
When we factor [tex]\(2a^2b^2c^3\)[/tex] out of the entire expression, we get:
[tex]\[
2a^4b^2c^3 + 8a^2b^3c^4 + 6a^4b^3c^3 = 2a^2b^2c^3 (a^2 + 4bc + 3a^2b)
\][/tex]
### Step 3: Simplify the Remaining Expression
Combine the terms inside the parentheses:
[tex]\[
a^2 + 4bc + 3a^2b
\][/tex]
### Step 4: Arriving at the Final Expression
We can combine these terms directly as they are already in a simplified form:
So the completely factorized form is:
[tex]\[
2a^2b^2c^3(a^2 + 4bc + 3a^2b)
\][/tex]
Thus, the completely factorized form of [tex]\(2a^4b^2c^3 + 8a^2b^3c^4 + 6a^4b^3c^3\)[/tex] is [tex]\(\boxed{2a^2b^2c^3(3a^2b + a^2 + 4bc)}\)[/tex].
