The problem asks us to complete the square for the quadratic expression $3x^2 + 9x + 4$.

1. Problem Description

The problem asks us to complete the square for the quadratic expression $3x^2 + 9x + 4$.

2. Solution Steps

First, factor out the coefficient of the $x^2$ term (which is 3) from the terms containing $x^2$ and $x$:

$3x^2 + 9x + 4 = 3(x^2 + 3x) + 4$

To complete the square for the expression inside the parentheses, $x^2 + 3x$, we need to add and subtract $(\frac{3}{2})^2 = \frac{9}{4}$ inside the parentheses:

$3(x^2 + 3x + \frac{9}{4} - \frac{9}{4}) + 4$

Now, rewrite the expression as:

$3((x + \frac{3}{2})^2 - \frac{9}{4}) + 4$

Distribute the 3:

$3(x + \frac{3}{2})^2 - 3(\frac{9}{4}) + 4$

$3(x + \frac{3}{2})^2 - \frac{27}{4} + \frac{16}{4}$

Combine the constant terms:

$3(x + \frac{3}{2})^2 - \frac{11}{4}$

3. Final Answer

$3(x + \frac{3}{2})^2 - \frac{11}{4}$