To solve the equation \( 4x^2 + 12x + 5 = -4 \), we first move the -4 to the left side:
\[ 4x^2 + 12x + 5 + 4 = 0 \]
This simplifies to:
\[ 4x^2 + 12x + 9 = 0 \]
Next, we can factor the quadratic equation. We look for two numbers that multiply to \( 4 \cdot 9 = 36 \) and add up to \( 12 \). The numbers 6 and 6 fit this requirement.
Thus, we can write the equation as:
\[ 4x^2 + 6x + 6x + 9 = 0 \]
Now, we group the terms:
\[ (4x^2 + 6x) + (6x + 9) = 0 \]
Factoring by grouping gives:
\[ 2x(2x + 3) + 3(2x + 3) = 0 \]
Now factor out the common term \( (2x + 3) \):
\[ (2x + 3)(2x + 3) = 0 \] or simplifying further, \[ (2x + 3)^2 = 0 \]
Now we can set the factor equal to zero:
\[ 2x + 3 = 0 \]
Solving for \( x \):
\[ 2x = -3 \] \[ x = -\frac{3}{2} \]
Thus, the solution to the equation is:
\[ x = -\frac{3}{2} \]
In terms of your options, that is equivalent to:
\[ x = -2 \frac{1}{2} \quad \text{or} \quad x = -\frac{3}{2} \]
So the answer is \( x = -2 \frac{1}{2} \) or negative \( \frac{3}{2} \).