To determine which expression is equivalent to [tex]\( 6x^8 y^2 + 12x^2 y^2 \)[/tex], we will proceed by factoring out the greatest common factor (GCF) from the terms of the expression.
1. Identify the GCF:
- The coefficients of the terms are 6 and 12.
- The GCF of 6 and 12 is 6.
- The variable terms include [tex]\( x^8 \)[/tex] and [tex]\( x^2 \)[/tex].
- The GCF of [tex]\( x^8 \)[/tex] and [tex]\( x^2 \)[/tex] is [tex]\( x^2 \)[/tex] (since [tex]\( x^2 \)[/tex] is the highest power of [tex]\( x \)[/tex] that divides both [tex]\( x^8 \)[/tex] and [tex]\( x^2 \)[/tex]).
- The variable terms include [tex]\( y^2 \)[/tex] in both terms.
- The GCF of [tex]\( y^2 \)[/tex] and [tex]\( y^2 \)[/tex] is [tex]\( y^2 \)[/tex] (since [tex]\( y^2 \)[/tex] is common to both terms).
Thus, the GCF of the entire expression is [tex]\( 6x^2 y^2 \)[/tex].
2. Factor out the GCF:
- Starting with [tex]\( 6x^8 y^2 + 12x^2 y^2 \)[/tex]:
- [tex]\( 6x^8 y^2 = 6x^2 y^2 \cdot x^6 \)[/tex]
- [tex]\( 12x^2 y^2 = 6x^2 y^2 \cdot 2 \)[/tex]
- Therefore, we can write the expression as:
[tex]\[
6x^8 y^2 + 12x^2 y^2 = 6x^2 y^2 (x^6) + 6x^2 y^2 (2)
\][/tex]
3. Combine the factored terms:
- Combine the factored form into a single expression:
[tex]\[
6x^2 y^2 (x^6 + 2)
\][/tex]
So, the expression [tex]\( 6x^2 y^2 (x^6 + 2) \)[/tex] is equivalent to [tex]\( 6x^8 y^2 + 12x^2 y^2 \)[/tex].
Therefore, the correct choice is:
C) [tex]\( 6 x^2 y^2 \left( x^6 + 2 \right) \)[/tex].
