To factor the polynomial expression [tex]\( 6a^2 + 6ca - 12a - 12c \)[/tex] completely, we should follow these detailed steps:
1. Group the terms:
Group the terms in pairs to facilitate factoring by grouping.
[tex]\[ 6a^2 + 6ca - 12a - 12c \][/tex]
2. Factor out the greatest common factor (GCF) from each pair:
- From the first pair [tex]\( 6a^2 + 6ca \)[/tex], the GCF is [tex]\( 6a \)[/tex]. Factoring out [tex]\( 6a \)[/tex] gives:
[tex]\[ 6a(a + c) \][/tex]
- From the second pair [tex]\( -12a - 12c \)[/tex], the GCF is [tex]\( -12 \)[/tex]. Factoring out [tex]\( -12 \)[/tex] gives:
[tex]\[ -12(a + c) \][/tex]
3. Factor by grouping:
Since both groups have a common factor of [tex]\( (a + c) \)[/tex], we can factor this common term out:
[tex]\[ 6a(a + c) - 12(a + c) = (a + c)(6a - 12) \][/tex]
4. Factor the remaining quadratic term completely:
We can factor out the GCF of [tex]\( 6 \)[/tex] from [tex]\( 6a - 12 \)[/tex]:
[tex]\[ 6(a - 2) \][/tex]
Therefore, the expression becomes:
[tex]\[ (a + c)6(a - 2) \][/tex]
5. Write the expression in its fully factored form:
Multiply the constants together:
[tex]\[ 6(a - 2)(a + c) \][/tex]
So, the completely factored form of the polynomial [tex]\( 6a^2 + 6ca - 12a - 12c \)[/tex] is:
[tex]\[ 6(a - 2)(a + c) \][/tex]
Therefore, the correct answer is:
[tex]\[ 6(a - 2)(a + c) \][/tex]
