a) To factor the expression x^3 - 6x^2 + 12x - 7, we can use the Rational Root Theorem to find possible rational roots. The possible rational roots are the factors of the constant term (-7) divided by the factors of the leading coefficient (1).

The factors of -7 are ±1 and ±7, and the factors of 1 are ±1. Therefore, the possible rational roots are ±1 and ±7.

By trying these values, we find that x = 1 is a root of the equation. Therefore, (x - 1) is a factor of the expression.

Using polynomial long division or synthetic division, we can divide x^3 - 6x^2 + 12x - 7 by (x - 1) to find the other factor:

x^2 - 5x + 7
____________________
x - 1 | x^3 - 6x^2 + 12x - 7
- (x^3 - x^2)
_____________
-5x^2 + 12x
+ (5x^2 - 5x)
_____________
17x - 7
- (17x - 17)
_____________
10

Therefore, the factored form of x^3 - 6x^2 + 12x - 7 is (x - 1)(x^2 - 5x + 7).

b) To factor the expression 2x^3 + 6x^2 + 12x + 8, we can first factor out the greatest common factor, which is 2:

2(x^3 + 3x^2 + 6x + 4)

Now, we can try to find rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term (4) divided by the factors of the leading coefficient (1).

The factors of 4 are ±1, ±2, and ±4, and the factors of 1 are ±1. Therefore, the possible rational roots are ±1, ±2, and ±4.

By trying these values, we find that x = -1 is a root of the equation. Therefore, (x + 1) is a factor of the expression.

Using polynomial long division or synthetic division, we can divide x^3 + 3x^2 + 6x + 4 by (x + 1) to find the other factor:

x^2 + 2x + 4
____________________
x + 1 | x^3 + 3x^2 + 6x + 4
- (x^3 + x^2)
_____________
2x^2 + 6x
+ (2x^2 + 2x)
_____________
8x + 4
- (8x + 8)
_____________
-4

Therefore, the factored form of 2x^3 + 6x^2 + 12x + 8 is 2(x + 1)(x^2 + 2x + 4).