Let's first distribute and simplify the left side of the equation:

5x(x-3) - 2(x-7) + 7(2x+6)
= 5x^2 - 15x - 2x + 14 + 14x + 42
= 5x^2 - 15x - 2x + 14 + 14x + 42
= 5x^2 - 17x + 56

Now, set the equation equal to 7 and solve for x:

5x^2 - 17x + 56 = 7
5x^2 - 17x + 56 - 7 = 0
5x^2 - 17x + 49 = 0

Now, factor the quadratic equation:

5x^2 - 17x + 49 = (5x - 7)(x - 7) = 0

Now, set each factor equal to zero and solve for x:

5x - 7 = 0 or x - 7 = 0
5x = 7 or x = 7
x = 7/5 or x = 7

Therefore, the solutions to the equation are x = 7/5 or x = 7.