Let 𝛼, 𝛽   be the roots of the equation (𝑥 − 𝑎)(𝑥 − 𝑏) = 𝑐, 𝑐 ≠ 0 . Then the roots of the equation (𝑥 − 𝛼)(𝑥 − 𝛽) + 𝑐 = 0  are

$(x-a)(x-b)=c$

$x^2-bx-ax+ab-c=0$

$\alpha +\beta =a+b,\alpha \beta =ab-c$

$(x-\alpha )(x-\beta )+c=0$

$x^2-(a+b)x+ab-c=0$

$x^2-(\alpha +\beta )x+\alpha \beta +c=0$

$x^2-(a+b)x+ab-c+c=0$

$x^2-(a+b)x+ab=0$

$\therefore$roots are $a,b$