To solve the equation \( 4x^2 + 12x + 5 = -4 \), we first move all terms to one side of the equation to set it to zero:
\[ 4x^2 + 12x + 5 + 4 = 0 \]
This simplifies to:
\[ 4x^2 + 12x + 9 = 0 \]
Next, we can factor this quadratic equation. Notice that \( 4x^2 + 12x + 9 \) can be rewritten as:
\[ (2x + 3)(2x + 3) = 0 \]
or
\[ (2x + 3)^2 = 0 \]
Now, to solve for \( x \), we 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:
\[ x = -\frac{3}{2} \]
This corresponds with the option \( x = -1 \frac{1}{2} \) from your responses.
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