To factor the quadratic expression \(2u^2 - 13u + 11\), we can use the method of finding two numbers that multiply to give the product of the leading coefficient (2) and the constant term (11), which is \(2 \cdot 11 = 22\), and add up to the middle coefficient (-13).
We need two numbers that multiply to 22 and add to -13. The numbers that satisfy these conditions are -11 and -2.
Now we can rewrite the middle term (-13u) using these two numbers:
\[ 2u^2 - 11u - 2u + 11 \]
Next, we can group the terms:
\[ (2u^2 - 11u) + (-2u + 11) \]
Factoring by grouping gives us:
\[ u(2u - 11) - 1(2u - 11) \]
Now, we can factor out the common factor \((2u - 11)\):
\[ (2u - 11)(u - 1) \]
Therefore, the complete factorization of \(2u^2 - 13u + 11\) is:
\[ (2u - 11)(u - 1) \]