To factor the quadratic expression \(3k^2 - 10k + 8\), we can start by looking for two numbers that multiply to \(3 \times 8 = 24\) (the product of the leading coefficient and the constant term) and that add up to \(-10\) (the coefficient of the middle term).
The factors of \(24\) that add up to \(-10\) are \(-6\) and \(-4\).
Next, we can rewrite the middle term \(-10k\) using \(-6k\) and \(-4k\):
\[ 3k^2 - 6k - 4k + 8 \]
Now, we can group the terms:
\[ (3k^2 - 6k) + (-4k + 8) \]
Factoring out the common factors from each group gives us:
\[ 3k(k - 2) - 4(k - 2) \]
Now, we can factor out the common binomial factor \((k - 2)\):
\[ (3k - 4)(k - 2) \]
Thus, the completely factored form of the quadratic \(3k^2 - 10k + 8\) is:
\[ (3k - 4)(k - 2) \]