$$f(x+h) =(b - A(x+h))^T (b - A(x+h)) + \alpha (x+h)^T (x+h) = (b - Ax - Ah)^T (b - Ax - Ah) + \alpha (x+h)^T ( x +h)$$ $\text{Letting } u = b - Ax \\ v = Ah Then F(x+h) = (u - v)^T (u - v) + \alpha (x+h)^T ( x +h) $\\ \\* $ F(x+h) = u^T u - v^T u - u^T v + v^T v + \alpha \left [ x^T x + x^T h + h^T x + h^T h \right ]$\\ \\* $ F(x) = (b - Ax)^T(b - Ax) + \alpha x^T x$\\ \\* $F(x+h) - F(x) = - v^T u - u^T v + v^T v + \alpha \left [ x^T x + x^T h + h^T x + h^T h \right ] - \alpha x^T x$\\ \\* $F(x+h) - F(x) = -2 v^T u + v^T v + \alpha [2 h^T x + h^T h]$