Answer
$$(x+3)^2+(y-2)^2+(z-4)^2=9$$
Work Step by Step
Center is $(-3,2,4)$ and tangent to $yz$. If it is tangent to the $yz$-plane, then the distance from the surface to the center is just the magnitude of the $x$-coordinate of the center. Therefore, the radius is $3$. So the equation of the sphere is
$$(x+3)^2+(y-2)^2+(z-4)^2=9$$
