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

1 + 45/(y^2 - 8y + 16) = 14/(y - 4)

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 fractions on the left side:

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:

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

Now, we can set the left side equal to the right side:

(y^2 - 8y + 61)/((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^2 - 8y + 61) = 14 * (y - 4)

Expanding the right side:

(y^2 - 8y + 61) = 14y - 56

Rearranging the equation:

y^2 - 8y + 61 - 14y + 56 = 0

Combining like terms:

y^2 - 22y + 117 = 0

Now, we can solve this quadratic equation. However, it cannot be factored easily, so we can use the quadratic formula:

y = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = -22, and c = 117:

y = (-(-22) ± √((-22)^2 - 4(1)(117))) / (2(1))

Simplifying:

y = (22 ± √(484 - 468)) / 2

y = (22 ± √16) / 2

y = (22 ± 4) / 2

Now, we have two possible solutions:

y = (22 + 4) / 2 = 26 / 2 = 13

y = (22 - 4) / 2 = 18 / 2 = 9

Therefore, the solutions to the equation are y = 13 and y = 9.