Start with the inequality:

\(2x^2 - 6x + 5 > 13\)

Simplify the inequality:

\(2x^2 - 6x + 5 - 13 > 0\)

\(2x^2 - 6x - 8 > 0\)

Factor the quadratic:

\(2(x^2 - 3x - 4) > 0\)

\(2(x - 4)(x + 1) > 0\)

Find the critical points by setting the factors to zero:

\(x - 4 = 0 \Rightarrow x = 4\)

\(x + 1 = 0 \Rightarrow x = -1\)

Test intervals around the critical points:

• For \(x < -1\), choose \(x = -2\): \(2(-2 - 4)(-2 + 1) > 0 \Rightarrow 2(-6)(-1) > 0 \Rightarrow 12 > 0\)
• For \(-1 < x < 4\), choose \(x = 0\): \(2(0 - 4)(0 + 1) > 0 \Rightarrow 2(-4)(1) > 0 \Rightarrow -8 > 0\) (false)
• For \(x > 4\), choose \(x = 5\): \(2(5 - 4)(5 + 1) > 0 \Rightarrow 2(1)(6) > 0 \Rightarrow 12 > 0\)

Thus, the solution is \(x > 4\) or \(x < -1\).