`x^2+6x+4`
Which of the following is equivalent to the expression above ?

Options

A. `(x+3)^2+5`
B. `(x+3)^2 -5`
C. `(x-3)^2+5`
D. `(x-3)^2-5`

Explanation

To solve the inequality \( x^2 + 6x + 4 < 0 \) and find an equivalent expression, we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x^2 + 6x + 4 \]

### Step 2: Complete the square
To complete the square, we need to express the quadratic in the form \( (x + a)^2 + b \).

1. Identify the coefficient of \( x \), which is 6. Half of this value is 3.
2. Square this value: \( 3^2 = 9 \).

Now, we can rewrite the expression by adding and subtracting 9:
\[ x^2 + 6x + 9 - 9 + 4 \]
This simplifies to:
\[ (x + 3)^2 - 5 \]

### Step 3: Final expression
Thus, we can rewrite the original expression as:
\[ (x + 3)^2 - 5 \]

### Conclusion
The equivalent expression to \( x^2 + 6x + 4 \) is:
\[ (x + 3)^2 - 5 \]