a) To solve the equation x^2 - 4x + 5(4 - x) = 0, we can simplify it first:

x^2 - 4x + 20 - 5x = 0
x^2 - 9x + 20 = 0

Now, we can factorize the quadratic equation:

(x - 4)(x - 5) = 0

Setting each factor equal to zero, we have:

x - 4 = 0 or x - 5 = 0

Solving for x in each equation, we get:

x = 4 or x = 5

Therefore, the solutions to the equation are x = 4 and x = 5.

b) To solve the equation (x - 7)(x^2 + x + 6) = ^2 - 7x, we can simplify it first:

(x - 7)(x^2 + x + 6) = x^2 - 7x

Expanding the left side of the equation, we have:

x^3 + x^2 + 6x - 7x^2 - 7x - 42 = x^2 - 7x

Combining like terms, we get:

x^3 - 6x^2 - 14x - 42 = x^2 - 7x

Moving all terms to one side of the equation, we have:

x^3 - 7x^2 + 7x - 42 = 0

Unfortunately, this equation cannot be easily factored or solved algebraically. You may need to use numerical methods or a graphing calculator to find the approximate solutions.