2x² - 6x + 8

Step 1: Identify the given expression. The expression to be factored is $2x^2 - 6x + 8$.

Step 2: Find the greatest common factor (GCF) of the terms. The coefficients are 2, -6, and 8. The greatest common factor of these numbers is 2. There is no common variable factor since the last term (8) does not have $x$.

Step 3: Factor out the GCF from the expression. Divide each term by 2: $2x^2 \div 2 = x^2$ $-6x \div 2 = -3x$ $8 \div 2 = 4$ So, the expression becomes $2(x^2 - 3x + 4)$.

Step 4: Attempt to factor the quadratic trinomial inside the parenthesis, $x^2 - 3x + 4$. To factor this trinomial, we look for two numbers that multiply to $c=4$ and add up to $b=-3$. The pairs of factors for 4 are (1, 4), (-1, -4), (2, 2), (-2, -2). Their sums are: $1+4 = 5$ $-1+(-4) = -5$ $2+2 = 4$ $-2+(-2) = -4$ None of these pairs sum to -3. Therefore, the quadratic trinomial $x^2 - 3x + 4$ cannot be factored further into linear factors with real coefficients.

The fully factored form of the expression is $2(x^2 - 3x + 4)$.

$\boxed{2(x^2 - 3x + 4)}$