To factor the polynomial expression [tex]\(6 a^2 + 6 c a - 12 a - 12 c\)[/tex] completely, we can follow these steps:
1. Group the terms:
Group the polynomial into pairs of terms that have common factors:
[tex]\[
6 a^2 + 6 c a - 12 a - 12 c = (6 a^2 + 6 c a) + (-12 a - 12 c)
\][/tex]
2. Factor out the greatest common factor from each group:
For the first group [tex]\((6 a^2 + 6 c a)\)[/tex], factor out the common factor [tex]\(6 a\)[/tex]:
[tex]\[
6 a (a + c)
\][/tex]
For the second group [tex]\((-12 a - 12 c)\)[/tex], factor out the common factor [tex]\(-12\)[/tex]:
[tex]\[
-12(a + c)
\][/tex]
3. Rewrite the grouped terms with their factored forms:
Now substitute the factored forms back into the expression:
[tex]\[
6 a (a + c) - 12(a + c)
\][/tex]
4. Factor out the common binomial factor [tex]\((a + c)\)[/tex]:
Notice that [tex]\((a + c)\)[/tex] is a common factor in both terms:
[tex]\[
6 a (a + c) - 12(a + c) = (a + c)(6 a - 12)
\][/tex]
5. Simplify the remaining factor:
Finally, factor out the constant term from the remaining factor if possible:
[tex]\[
6 a - 12 = 6(a - 2)
\][/tex]
Substitute this back into the expression to get the completely factored form:
[tex]\[
(a + c) [6 (a - 2)]
\][/tex]
6. Combine the factors:
Rearrange the binomials to present the final factored expression:
[tex]\[
6 (a - 2) (a + c)
\][/tex]
Thus, the completely factored form of the polynomial [tex]\(6 a^2 + 6 c a - 12 a - 12 c\)[/tex] is:
[tex]\[
\boxed{6 (a - 2) (a + c)}
\][/tex]
