1. The problem is to rewrite the quadratic function $$f(x) = x^{2} + 16x - 46$$ by completing the square. 2. The formula for completing the square for a quadratic $$ax^{2} + bx + c$$ is to write it as $$a(x-h)^{2} + k$$ where $$h = -\frac{b}{2a}$$ and $$k$$ is the value of the function at $$x = h$$. 3. For the given function, $$a = 1$$, $$b = 16$$, and $$c = -46$$. 4. Calculate $$h$$: $$h = -\frac{16}{2 \times 1} = -8$$ 5. Rewrite the function by adding and subtracting $$h^{2} = (-8)^{2} = 64$$ inside the expression: $$f(x) = x^{2} + 16x - 46 = (x^{2} + 16x + 64) - 64 - 46$$ 6. Factor the perfect square trinomial: $$f(x) = (x + 8)^{2} - 110$$ 7. So, the function rewritten by completing the square is: $$f(x) = (x + 8)^{2} - 110$$