To solve this equation, we can use the properties of logarithms.

First, we can combine the two logarithms on the left side of the equation using the product rule of logarithms, which states that log_a(x) + log_a(y) = log_a(xy).

So, we have:

log7((x-4)(x+1)) = log7(4x+4)

Next, we can simplify the expression on the left side:

(x-4)(x+1) = 4x + 4

Expanding the left side:

x^2 - 3x - 4 = 4x + 4

Rearranging the equation:

x^2 - 7x - 8 = 0

Now, we can factor the quadratic equation:

(x - 8)(x + 1) = 0

Setting each factor to zero:

x - 8 = 0 --> x = 8
x + 1 = 0 --> x = -1

Therefore, the solutions to the equation are x = 8 and x = -1.