1. **State the problem:** Convert the quadratic equation $y = x^2 + 10x + 22$ into vertex form. 2. **Recall the vertex form:** The vertex form of a quadratic is $$y = a(x-h)^2 + k$$ where $(h,k)$ is the vertex. 3. **Complete the square:** Start with the given equation: $$y = x^2 + 10x + 22$$ Group the $x$ terms: $$y = (x^2 + 10x) + 22$$ To complete the square, take half of 10, which is 5, and square it: $5^2 = 25$. Add and subtract 25 inside the parentheses: $$y = (x^2 + 10x + 25 - 25) + 22$$ Rewrite: $$y = (x^2 + 10x + 25) - 25 + 22$$ Simplify constants: $$y = (x + 5)^2 - 3$$ 4. **Interpretation:** The vertex is at $(-5, -3)$, and the parabola opens upwards because the coefficient of $(x+5)^2$ is positive. **Final answer:** $$y = (x + 5)^2 - 3$$