To determine which binomial is a factor of the given quadratic expression, we can use the factor theorem. According to the factor theorem, if a polynomial expression P(x) is divided by (x - a) and the remainder is 0, then (x - a) is a factor of P(x).

In this case, the given quadratic expression is x^2 - 6x + 8. We need to find which binomial is a factor of this expression.

Let's test each option to see which one satisfies the factor theorem.

Option A: (x - 4)
To determine if (x - 4) is a factor, we divide the given expression by (x - 4) using polynomial long division:

x - 2
__________________________
x - 4 | x^2 - 6x + 8
- (x^2 - 4x)
_____________
-2x + 8
-(-2x + 8)
___________
0

Since the remainder is 0, we can conclude that (x - 4) is a factor of x^2 - 6x + 8.

Option B: (x + 4)
Dividing x^2 - 6x + 8 by (x + 4) using polynomial long division, we get:

x - 10
__________________________
x + 4 | x^2 - 6x + 8
- (x^2 + 4x)
_____________
-10x + 8
-(-10x - 40)
____________
48

The remainder is not zero, so (x + 4) is not a factor of x^2 - 6x + 8.

Option C: (x + 2)
Dividing x^2 - 6x + 8 by (x + 2) using polynomial long division, we get:

x - 8
__________________________
x + 2 | x^2 - 6x + 8
- (x^2 + 2x)
_____________
-8x + 8
-(-8x - 16)
____________
24

The remainder is not zero, so (x + 2) is not a factor of x^2 - 6x + 8.

Option D: (x - 8)
Dividing x^2 - 6x + 8 by (x - 8) using polynomial long division, we get:

x + 2
__________________________
x - 8 | x^2 - 6x + 8
- (x^2 - 8x)
_____________
2x + 8
-(2x - 16)
___________
24

The remainder is not zero, so (x - 8) is not a factor of x^2 - 6x + 8.

Thus, the only binomial that is a factor of x^2 - 6x + 8 is (x - 4), which corresponds to option A.