1. **State the problem:** Express the quadratic expression $7 - 12x + x^2$ in the form $p + (q + x)^2$, where $p$ and $q$ are integers. 2. **Recall the formula:** The expression $(q + x)^2$ expands to $x^2 + 2qx + q^2$. 3. **Match terms:** We want to rewrite $7 - 12x + x^2$ as $p + (q + x)^2 = p + x^2 + 2qx + q^2$. 4. **Compare coefficients:** - Coefficient of $x^2$ is $1$ on both sides, so no change needed. - Coefficient of $x$ is $-12$ on the left and $2q$ on the right, so $2q = -12$ which gives $q = -6$. 5. **Find $p$:** Substitute $q = -6$ into $p + q^2 = 7$: $$p + (-6)^2 = 7$$ $$p + 36 = 7$$ $$p = 7 - 36 = -29$$ 6. **Final expression:** $$7 - 12x + x^2 = -29 + (x - 6)^2$$ This completes the rewriting in the desired form.