Complete the square to re-write the quadratic function in vertex form: y = x^2 + 3x + 4

Steps to Solve

1. Identify the quadratic expression
   The given function is $y = x^2 + 3x + 4$. We need to focus on the terms $x^2 + 3x$ to complete the square.

2. Factor out the coefficient of $x^2$ (if necessary)
   In this case, the coefficient of $x^2$ is 1, so we can proceed without factoring.

3. Find the term to complete the square
   To find the term that completes the square, take half of the coefficient of $x$ (which is 3), square it, and add/subtract that value.
   Half of 3 is $\frac{3}{2}$, and squaring it gives:
   $$ \left(\frac{3}{2}\right)^2 = \frac{9}{4} $$

4. Rewrite the quadratic expression
   Now, we can write the function as:
   $$ y = (x^2 + 3x + \frac{9}{4}) + 4 - \frac{9}{4} $$
   This simplifies to:
   $$ y = \left(x + \frac{3}{2}\right)^2 + \frac{16}{4} - \frac{9}{4} $$

5. Simplify the constant terms
   Now combine the constants:
   $$ \frac{16}{4} - \frac{9}{4} = \frac{7}{4} $$
   Thus, the function becomes:
   $$ y = \left(x + \frac{3}{2}\right)^2 + \frac{7}{4} $$

6. Write the final vertex form
   The vertex form of the quadratic function is:
   $$ y = \left(x + \frac{3}{2}\right)^2 + \frac{7}{4} $$
   where the vertex is at $\left(-\frac{3}{2}, \frac{7}{4}\right)$.