α and β are the zeroes of $x^2+5x+4$. Form the quadratic polynomial (leading coefficient 1) whose zeroes are α + 3 and β + 3.

$\alpha +\beta =-5$ and $\alpha \beta =4$. New sum $=(\alpha +3)+(\beta +3)=-5+6=1$. New product $=(\alpha +3)(\beta +3)=\alpha \beta +3(\alpha +\beta )+9=4-15+9=-2$. So the polynomial is $x^2-(1)x+(-2)=x^2-x-2$. (Check: the zeroes −1 and −4 shift to 2 and −1.) $x^2+x-2$ flips the sign of the new sum; $x^2+11x+28$ shifts the zeroes by −3 instead of +3; $x^2-x+2$ keeps the product positive.