For this problem, we are going to use the Remainder Theorem. This says that for [tex]x - n[/tex] to be a factor of a polynomial [tex]p(x)[/tex], then [tex]p(n) = 0[/tex]. Essentially, it says that [tex]x - n[/tex] is a factor if when you substitute [tex]n[/tex] into the polynomial you get a result of 0.
Thus, in our case, when we substitute [tex]x = 2[/tex] into the polynomial, we should get an answer of 0 if [tex]x - 2[/tex] is a factor of the polynomial. Given this information, we can solve for [tex]c[/tex]:
[tex]p(2) = 2^3 - 4(2^2) + 2c + 2 = 0[/tex]
[tex]8 - 16 + 2c + 2 = 0[/tex]
[tex]2c - 6 = 0[/tex]
[tex]2c = 6[/tex]
[tex]c = 3[/tex]
The solution is c = 3.
