1. **State the problem:** We need to find the area of triangle $\triangle PQR$ where side $r=4$ inches, side $p=8.4$ inches, and the included angle $\angle Q=174^\circ$. 2. **Formula used:** The area of a triangle given two sides and the included angle is $$\text{Area} = \frac{1}{2} \times a \times b \times \sin(C)$$ where $a$ and $b$ are the sides and $C$ is the included angle. 3. **Apply the formula:** Here, $a = p = 8.4$, $b = r = 4$, and $C = 174^\circ$. 4. Calculate the sine of the angle: $$\sin(174^\circ) = \sin(180^\circ - 6^\circ) = \sin(6^\circ) \approx 0.1045$$ 5. Substitute values: $$\text{Area} = \frac{1}{2} \times 8.4 \times 4 \times 0.1045$$ 6. Simplify step-by-step: $$= \frac{1}{2} \times 33.6 \times 0.1045$$ $$= 16.8 \times 0.1045$$ 7. Multiply: $$= 1.7556$$ 8. **Final answer:** Rounded to the nearest tenth, $$\boxed{1.8 \text{ square inches}}$$