1. **State the problem:** We are given a quadrilateral EBCD with angles at points B, C, and F. We know that $m\angle B = (14x + 4)^\circ$, $m\angle C = 51^\circ$, and $m\angle F = 55^\circ$. We need to solve for $x$. 2. **Identify the relationship:** Since F is the intersection of diagonals inside the quadrilateral, angles around point F sum to $360^\circ$. However, the problem focuses on $m\angle B$ and $m\angle C$ with $m\angle C = 51^\circ$ given. 3. **Assumption:** If $m\angle B$ and $m\angle C$ are adjacent angles on a straight line or part of a linear pair, their measures sum to $180^\circ$. 4. **Set up the equation:** $$ (14x + 4) + 51 = 180 $$ 5. **Solve for $x$:** $$ 14x + 4 + 51 = 180 $$ $$ 14x + 55 = 180 $$ $$ 14x = 180 - 55 $$ $$ 14x = 125 $$ $$ x = \frac{125}{14} $$ 6. **Simplify the fraction:** $$ x = 8.92857142857 \approx 8.93 $$ **Final answer:** $x \approx 8.93$