To determine which value of [tex]\(c\)[/tex] would make the expression [tex]\(8x + cy\)[/tex] completely factored, we need to consider the greatest common divisor (GCD) of the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
The expression [tex]\(8x + cy\)[/tex] can be completely factored if the coefficient [tex]\(c\)[/tex] is a multiple of the coefficient of [tex]\(x\)[/tex], which is 8. This means [tex]\(c\)[/tex] must be a multiple of 8.
Let's evaluate each of the given options:
1. [tex]\( c = 2 \)[/tex]
- [tex]\( 2 \div 8 = 0.25 \)[/tex] (not a whole number)
2. [tex]\( c = 7 \)[/tex]
- [tex]\( 7 \div 8 = 0.875 \)[/tex] (not a whole number)
3. [tex]\( c = 12 \)[/tex]
- [tex]\( 12 \div 8 = 1.5 \)[/tex] (not a whole number)
4. [tex]\( c = 16 \)[/tex]
- [tex]\( 16 \div 8 = 2 \)[/tex] (a whole number)
Since 16 is the only value among the given options that is a multiple of 8, the expression [tex]\(8x + 16y\)[/tex] is completely factored because both terms share a common factor of 8.
Thus, the value of [tex]\(c\)[/tex] that makes the expression [tex]\(8x + cy\)[/tex] completely factored is:
[tex]\[
\boxed{16}
\][/tex]
