To find the equation of the plane α that passes through the point M(1,2,1) and is parallel to the plane β with equation 4x - y + z + 10 = 0, we need to find a normal vector for plane α.
Since plane α is parallel to plane β, the normal vector of plane α will also be the normal vector of plane β.
The normal vector of plane β can be found by looking at the coefficients of x, y, and z in the equation of plane β.
The coefficients of x, y, and z in the equation 4x - y + z + 10 = 0 are 4, -1, and 1 respectively.
Thus, the normal vector of plane α is N(4, -1, 1).
Now, we can use the point-normal form of the equation of a plane to find the equation of plane α.
The point-normal form of the equation of a plane is given by:
N * (r - r0) = 0,
where N is the normal vector of the plane,
r is any point on the plane, and
r0 is a specific point on the plane.
In this case, we can use the point M(1,2,1) on plane α and the normal vector N(4, -1, 1) to write the equation of plane α as:
(4, -1, 1) * (r - (1,2,1)) = 0.
Expanding this equation gives:
4(x - 1) - (y - 2) + (z - 1) = 0.
Simplifying further:
4x - 4 - (y - 2) + z - 1 = 0.
Combining like terms:
4x - y + z - 3 = 0.
Therefore, the equation of plane α is:
\boxed{4x - y + z - 3 = 0}.