To solve the equation \( 4x^2 + 12x + 5 = -4 \), we first need to rearrange it into standard quadratic form:
\[ 4x^2 + 12x + 5 + 4 = 0 \]
This simplifies to:
\[ 4x^2 + 12x + 9 = 0 \]
Now, we can factor this quadratic equation. We notice that \( 4x^2 + 12x + 9 \) can be factored as:
\[ (2x + 3)(2x + 3) = (2x + 3)^2 \]
Setting this equal to zero gives us:
\[ (2x + 3)^2 = 0 \]
Taking the square root of both sides:
\[ 2x + 3 = 0 \]
Now, solve for \( x \):
\[ 2x = -3 \] \[ x = -\frac{3}{2} \]
Now let’s compare this with the potential answers provided:
- \( x = \frac{23}{3} \)
- \( x = 1\frac{1}{2} \) (which is \( \frac{3}{2} \))
- \( x = -1\frac{1}{2} \) (which is \( -\frac{3}{2} \))
- \( x = -\frac{2}{3} \)
The correct answer is:
\[ x = -1\frac{1}{2} \quad \text{(which is } -\frac{3}{2} \text{)} \]