$\displaystyle 6(x-3)^2+13(x-3)+5$
Use a dummy variable $\displaystyle X = x - 3$ so that the expression becomes
$\displaystyle 6X^2 + 13X + 5$
$\displaystyle = 6X^2 + 3X + 10X + 5$
$\displaystyle = 3X(2X + 1) + 5(2X + 1)$
$\displaystyle = (2X + 1)(3X + 5)$
And since $\displaystyle X = x - 3$, this becomes
$\displaystyle [2(x - 3) + 1][3(x - 3) + 5]$
$\displaystyle = (2x - 6 + 1)(3x - 9 + 5)$
$\displaystyle = (2x - 5)(3x - 4)$.