To solve the equation \(4x^2 + 12x + 5 = -4\), we first move all terms to one side of the equation:
\[ 4x^2 + 12x + 5 + 4 = 0 \]
This simplifies to:
\[ 4x^2 + 12x + 9 = 0 \]
Next, we can factor the quadratic. We look for two numbers that multiply to \(4 \times 9 = 36\) and add up to \(12\). The numbers \(6\) and \(6\) satisfy this condition, so we can write:
\[ 4x^2 + 6x + 6x + 9 = 0 \]
This can be factored as:
\[ (2x + 3)(2x + 3) = 0 \quad \text{or} \quad (2x + 3)^2 = 0 \]
Setting each factor to zero gives us:
\[ 2x + 3 = 0 \]
Solving for \(x\):
\[ 2x = -3 \implies x = -\frac{3}{2} \]
Thus, the solution to the equation \(4x^2 + 12x + 5 = -4\) is:
\[ x = -\frac{3}{2} \]
Among the response options, \( x = -\frac{3}{2} \) can be expressed as \(-112\) in fractional notation. Therefore, the correct response is:
\[ x = -\frac{3}{2} \] (or \(x = -112\))
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