1. **Problem statement:** We have a right-angled triangle with points R, T, Q, and S. Segments RS and SQ are given as expressions in terms of $x$: $RS = 3x + 2$ and $SQ = 5x - 8$. We need to find the length of segment $QS$. 2. **Understanding the problem:** Since $QS$ is the segment from $Q$ to $S$, and $RS$ and $SQ$ are parts of the triangle, we need to clarify if $QS$ is the sum of $RS$ and $SQ$ or if $QS$ is the segment labeled $5x - 8$. The problem asks "What is QS?" and gives options 2, 5, 17, 33 units. 3. **Assuming $QS$ is the sum of $RS$ and $SQ$:** $$QS = RS + SQ = (3x + 2) + (5x - 8) = 8x - 6$$ 4. **Finding $x$:** Since the triangle is right-angled and $RT$ is horizontal with equal segments, we can infer that $RS$ and $SQ$ relate to the sides of the triangle. However, the problem does not provide an explicit equation for $x$. We must use the right angle property or given equal segments to find $x$. 5. **Using the right angle at $T$ and equal segments on $RT$:** If $RT$ is divided into equal segments, and $RS$ and $SQ$ are legs of the triangle, then by the Pythagorean theorem: $$RS^2 + SQ^2 = RQ^2$$ But $RQ$ is not given. Without more information, we cannot solve for $x$ directly. 6. **Alternative approach:** The problem likely expects us to find $x$ such that $QS$ is a positive length matching one of the options. 7. **Check options by substituting $x$ values:** Try to find $x$ such that $QS = 5x - 8$ equals one of the options. Try $QS = 17$: $$17 = 5x - 8 \implies 5x = 25 \implies x = 5$$ Check $RS$ with $x=5$: $$RS = 3(5) + 2 = 15 + 2 = 17$$ So $RS = 17$ and $SQ = 17$. 8. **Conclusion:** Since $RS = SQ = 17$, $QS$ is $17$ units. **Final answer:** $$\boxed{17}$$