There are different methods to find the angle between two planes, but one common approach is to use the dot product. The normal vectors of two planes are perpendicular to them, so their dot product is zero. Therefore, if we can find the normal vectors of the two planes, we can use the dot product formula to find the cosine of the angle between them, and then take the inverse cosine to get the angle itself.

For the first plane, x + 2y + 3z + 1 = 0, we can read off the coefficients of x, y, and z to get the normal vector N1 = (1, 2, 3). Note that the magnitude of this vector is sqrt(1^2 + 2^2 + 3^2) = sqrt(14), but we don't need to normalize it for the dot product.

For the second plane, we need to use the given point (4, 1, -7) to find the normal vector. A normal vector is perpendicular to any vector lying in the plane, so we can subtract the point from a general point on the plane and then take the cross product of the resulting vectors. For example, if we let x, y, and z be arbitrary coordinates in the plane, then the vector from (4, 1, -7) to (x, y, z) is (x-4, y-1, z+7). We want this vector to be perpendicular to the given vector (4, 1, -7), so their dot product must be zero:

(4, 1, -7) dot (x-4, y-1, z+7) = 4(x-4) + (y-1) + (-7)(z+7) = 4x + y - 7z - 60 = 0

This is the equation of the plane, but we don't need it explicitly. Instead, we can read off the coefficients of x, y, and z to get the normal vector N2 = (4, 1, -7). Again, we don't need to normalize this vector for the dot product.

Now we can use the dot product formula:

N1 dot N2 = |N1| |N2| cos(theta)

where theta is the angle between the planes. Substituting the values we found, we get:

(1, 2, 3) dot (4, 1, -7) = sqrt(14) sqrt(66) cos(theta)

4 + 2 - 21 = sqrt(14) sqrt(66) cos(theta)

-15 = sqrt(924) cos(theta)

cos(theta) = -15/sqrt(924) = -5/22

Taking the inverse cosine (or arccosine) of this value, we get:

theta = 107.51 degrees

This is not one of the answer choices, but if we take the complementary angle (i.e. the angle between the normal vectors), we get:

theta' = 180 - 107.51 = 72.49 degrees

This is closest to answer choice (c) 11.23, but it is not exactly the same. However, it is possible that there is a mistake in the problem statement or the answer choices, or that we made an error in the calculation.