The expression you provided, $2x - 4x + 5$, seems to have a typo. The like terms should be combined. Assuming you mean $2x^2 - 4x + 5$, let's convert this quadratic expression into vertex form.

The vertex form of a quadratic equation is:

$y = a(x - h)^2 + k$

Where $(h, k)$ is the vertex of the parabola. Let's go step by step to convert $2x^2 - 4x + 5$ into vertex form.

Step 1: Factor out the coefficient of $x^2$ (if necessary)

Start by factoring out the leading coefficient $a = 2$ from the first two terms.

$y = 2(x^2 - 2x) + 5$

Step 2: Complete the square

Now we need to complete the square for the expression inside the parentheses, $x^2 - 2x$.

1. Take the coefficient of $x$, divide it by 2, and square it.

   The coefficient of $x$ is $-2$, so: $\left(\frac{-2}{2}\right)^2 = 1$

2. Add and subtract this value inside the parentheses. This allows us to complete the square: $y = 2(x^2 - 2x + 1 - 1) + 5$

   Simplifying: $y = 2((x - 1)^2 - 1) + 5$

Step 3: Simplify the expression

Now distribute the $2$ across the terms in the parentheses:

$y = 2(x - 1)^2 - 2 + 5$

Finally, combine the constants:

$y = 2(x - 1)^2 + 3$

Step 4: Identify the vertex

The equation is now in vertex form: $y = 2(x - 1)^2 + 3$

So the vertex of the parabola is $(h, k) = (1, 3)$.

---

Would you like to see more details or have any questions?

Here are 5 related questions to expand on this concept:

1. How do you find the axis of symmetry from the vertex form?
2. Can you graph a quadratic function given in vertex form?
3. What happens to the graph if the value of $a$ is negative?
4. How does completing the square help in solving quadratic equations?
5. What is the connection between vertex form and standard form of a quadratic equation?

Tip: Always ensure that when completing the square, you balance the equation by adding and subtracting the same value inside the parentheses.