The equation of the circle concentric with circle $x^2 + y^2 - 6x + 7 = 0$ and which touches the line $x + y + 3 = 0$ is ....

The given circle is $x^2 + y^2 - 6x + 7 = 0$.

Its center is $(3, 0)$.

The required circle is concentric with the given circle, so its center is also $(3, 0)$.

The required circle touches the line $x + y + 3 = 0$.

The radius $r$ of the required circle is the perpendicular distance from the center $(3, 0)$ to the line $x + y + 3 = 0$.

$r = \dfrac{|3 + 0 + 3|}{\sqrt{1^2 + 1^2}} = \dfrac{6}{\sqrt{2}} = 3\sqrt{2}$

The equation of the required circle is $(x - 3)^2 + (y - 0)^2 = r^2$.

$(x - 3)^2 + y^2 = (3\sqrt{2})^2$

$x^2 - 6x + 9 + y^2 = 18$

$x^2 + y^2 - 6x - 9 = 0$

Answer: $x^2 + y^2 - 6x - 9 = 0$