To express \(2x^2 - 10x + 8\) in the form \(a(x + b)^2 + c\), we complete the square:

Start with the quadratic expression:

\(2x^2 - 10x + 8\)

Factor out the 2 from the first two terms:

\(2(x^2 - 5x) + 8\)

Complete the square inside the parentheses:

\(x^2 - 5x = (x - \frac{5}{2})^2 - (\frac{5}{2})^2\)

\(= (x - \frac{5}{2})^2 - \frac{25}{4}\)

Substitute back into the expression:

\(2((x - \frac{5}{2})^2 - \frac{25}{4}) + 8\)

\(= 2(x - \frac{5}{2})^2 - \frac{25}{2} + 8\)

\(= 2(x - \frac{5}{2})^2 - \frac{25}{2} + \frac{16}{2}\)

\(= 2(x - \frac{5}{2})^2 - \frac{9}{2}\)

Thus, \(a = 2\), \(b = -\frac{5}{2}\), \(c = -\frac{9}{2}\).

The minimum value of \(2x^2 - 10x + 8\) occurs at the vertex of the parabola, which is \(-\frac{9}{2}\).