1. **State the problem:** Solve the equation $$16x + 16 = x^2 + 2x + 1$$. 2. **Rewrite the equation:** Notice that $$x^2 + 2x + 1 = (x + 1)^2$$, so the equation becomes: $$16x + 16 = (x + 1)^2$$ 3. **Bring all terms to one side:** $$0 = (x + 1)^2 - 16x - 16$$ Expand the square: $$0 = x^2 + 2x + 1 - 16x - 16$$ Simplify: $$0 = x^2 - 14x - 15$$ 4. **Solve the quadratic equation:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=1$$, $$b=-14$$, and $$c=-15$$. Calculate the discriminant: $$\Delta = (-14)^2 - 4 \times 1 \times (-15) = 196 + 60 = 256$$ Calculate the roots: $$x = \frac{-(-14) \pm \sqrt{256}}{2 \times 1} = \frac{14 \pm 16}{2}$$ 5. **Find the two solutions:** - For the plus sign: $$x = \frac{14 + 16}{2} = \frac{30}{2} = 15$$ - For the minus sign: $$x = \frac{14 - 16}{2} = \frac{-2}{2} = -1$$ 6. **Check for extraneous solutions:** Substitute $$x = -1$$ back into the original equation: $$16(-1) + 16 = -16 + 16 = 0$$ $$(-1)^2 + 2(-1) + 1 = 1 - 2 + 1 = 0$$ Both sides equal 0, so $$x = -1$$ is valid. Substitute $$x = 15$$: $$16(15) + 16 = 240 + 16 = 256$$ $$(15)^2 + 2(15) + 1 = 225 + 30 + 1 = 256$$ Both sides equal 256, so $$x = 15$$ is valid. **Final answer:** $$x = -1$$ or $$x = 15$$.