Find an equation of the plane α if it passes through the point M (1,2,1) and is parallel to the plane β with equation 4x - y + z + 10 = 0.

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}.