1. **State the problem:** Given $p=100000$ and the equation $p^2 + 2pq + q^2 = 1$, find the value of $q$. 2. **Recall the formula:** The equation $p^2 + 2pq + q^2 = 1$ can be recognized as $(p+q)^2 = 1$. 3. **Rewrite the equation:** $$p^2 + 2pq + q^2 = (p+q)^2 = 1$$ 4. **Solve for $q$:** Since $(p+q)^2 = 1$, then $$p + q = \pm 1$$ 5. **Isolate $q$:** $$q = \pm 1 - p$$ 6. **Substitute $p=100000$:** $$q = \pm 1 - 100000$$ 7. **Calculate the two possible values:** - For $q = 1 - 100000 = -99999$ - For $q = -1 - 100000 = -100001$ **Final answer:** $$q = -99999 \text{ or } q = -100001$$