The problem states that $R$ is the midpoint of $QS$. It provides the following information:

• $RS = 2x - 4$,
• $ST = 4x - 1$,
• $QT = 8x - 43$.

We are tasked with finding the length of $QS$.

Step-by-step solution:

1. Understand the midpoint condition: Since $R$ is the midpoint of $QS$, the length of $QS$ is twice the length of $RS$. This gives us: $QS = 2 \times RS = 2(2x - 4) = 4x - 8.$

2. Express $QT$ in terms of known lengths: The total length $QT$ is the sum of $QS$ and $ST$. So: $QT = QS + ST.$ Substitute the known expressions for $QT$, $QS$, and $ST$: $8x - 43 = (4x - 8) + (4x - 1).$

3. Simplify and solve for $x$: Combine like terms: $8x - 43 = 4x - 8 + 4x - 1,$ $8x - 43 = 8x - 9.$ Subtract $8x$ from both sides: $-43 = -9.$ This is clearly not possible, which suggests an error in my interpretation or setup. Let's reassess the problem.

Since the setup seems inconsistent, please let me double-check the conditions in case I misinterpreted the figure. Would you like me to try a different approach or check any specific step?