Using the completing-the-square method, rewrite f(x) = x2 + 10x + 7 in vertex form.

Let's begin with moving the constant to the side with f(x):
f(x) - 7 = x² + 10x

Next let's add (half of 10)² to both sides of the equation: btw that is 25 but on the right side we will leave it as 5² because it will make the factoring easier

f(x) - 7 + 25 = x² + 10x + 5²

Lets simplify the left side a bit and factor the perfect square trinomial on the right.

f(x) + 18 = (x + 5)²

Finally move the constant back to the side with the binomial

f(x) = (x + 5)² - 18

The vertex is (-5, -18)