Solve for \( h \). \[ 4 h^{2}+49 h+12=0 \]

We start with the quadratic equation \[ 4h^2 + 49h + 12 = 0. \] To solve for \( h \), we use the quadratic formula: \[ h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] where \( a = 4 \), \( b = 49 \), and \( c = 12 \). 1. Calculate the discriminant: \[ b^2 - 4ac = 49^2 - 4 \cdot 4 \cdot 12. \] Compute \( 49^2 \): \[ 49^2 = 2401. \] Then, compute \( 4 \cdot 4 \cdot 12 \): \[ 4 \cdot 4 \cdot 12 = 192. \] Thus, the discriminant is: \[ 2401 - 192 = 2209. \] 2. Find the square root of the discriminant: \[ \sqrt{2209} = 47, \] since \( 47^2 = 2209 \). 3. Substitute back into the quadratic formula: \[ h = \frac{-49 \pm 47}{2 \cdot 4} = \frac{-49 \pm 47}{8}. \] 4. Evaluate both cases: - For the positive case: \[ h = \frac{-49 + 47}{8} = \frac{-2}{8} = -\frac{1}{4}. \] - For the negative case: \[ h = \frac{-49 - 47}{8} = \frac{-96}{8} = -12. \] The solutions for \( h \) are: \[ h = -\frac{1}{4} \quad \text{or} \quad h = -12. \]