To solve the equation, we can start by simplifying both sides.

First, let's simplify the right side of the equation:

14/(y - 4) = 14/(y - 4)

Now, let's simplify the left side of the equation:

1 + 45/(y ^ 2 - 8y + 16) = 1 + 45/((y - 4)(y - 4))

Next, let's find the common denominator for the left side of the equation:

1 + 45/((y - 4)(y - 4)) = ((y - 4)(y - 4) + 45)/((y - 4)(y - 4))

Expanding the numerator on the left side:

((y - 4)(y - 4) + 45)/((y - 4)(y - 4)) = (y^2 - 8y + 16 + 45)/((y - 4)(y - 4))

Simplifying the numerator on the left side:

(y^2 - 8y + 16 + 45)/((y - 4)(y - 4)) = (y^2 - 8y + 61)/((y - 4)(y - 4))

Now, we have the equation:

((y - 4)(y - 4) + 45)/((y - 4)(y - 4)) = 14/(y - 4)

To eliminate the denominators, we can multiply both sides of the equation by ((y - 4)(y - 4)):

((y - 4)(y - 4) + 45)/((y - 4)(y - 4)) * ((y - 4)(y - 4)) = 14/(y - 4) * ((y - 4)(y - 4))

Simplifying the equation:

(y - 4)(y - 4) + 45 = 14

Expanding the left side:

y^2 - 8y + 16 + 45 = 14

Simplifying the equation:

y^2 - 8y + 61 = 14

Rearranging the equation:

y^2 - 8y + 47 = 0

Now, we have a quadratic equation. To solve it, we can use the quadratic formula:

y = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = -8, and c = 47. Substituting these values into the quadratic formula:

y = (-(-8) ± √((-8)^2 - 4(1)(47))) / (2(1))

Simplifying the equation:

y = (8 ± √(64 - 188)) / 2

y = (8 ± √(-124)) / 2

Since the square root of a negative number is not a real number, there are no real solutions to the equation.