Thinking mathematically and proving things I'm learning about existential and universal quantifiers and one of the examples given is: The equation $x^2 + 2x +1 = 0$ has a real root. Written using an existential quantifier it looks like this: $\exists x \mid x^2 + 2x + 1 = 0$ To prove the statement above, the lecturer chose the value $-1$ which satisfies the above and the result does indeed equal $0$. I come from a software development background so my obvious approach to finding that $x = -1$ would be to write a simple program to brute force it (which might not work). But as a mathematician how would you approach this (introspection would be really helpful for me)? Simply brute forcing this kind of things seems the wrong way to think since there are an infinite amount of numbers.

Just use the formula for finding solutions to quadratic equations, which are equations of the form $ax^2 + bx+c=0$:

$$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$

In your case, we have $a=1$, $b=2$, and $c=1$, so we get:

$$x = \frac{-2 \pm \sqrt{2^2-4}}{2}=\frac{-2 \pm \sqrt{0}}{2}=\frac{-2 }{2}=-1$$