If α and β are the roots of $x^2-4x+1=0$, then the equation whose roots are α + 3 and β + 3 is:

New sum = (α + β) + 6 = 4 + 6 = 10 and new product = αβ + 3(α + β) + 9 = 1 + 12 + 9 = 22, so the equation is $x^2-10x+22=0$. Equivalently, replace $x$ by $x-3$ in the original: $(x-3{)}^2-4(x-3)+1=x^2-10x+22$.