1. **State the problem:** Solve the quadratic equation $$4(x + 1)^2 - 7 = 45$$ for all values of $x$ in simplest form. 2. **Add 7 to both sides to isolate the squared term:** $$4(x + 1)^2 - 7 + 7 = 45 + 7$$ $$4(x + 1)^2 = 52$$ 3. **Divide both sides by 4 to solve for $(x + 1)^2$:** $$\frac{4(x + 1)^2}{\cancel{4}} = \frac{52}{\cancel{4}}$$ $$ (x + 1)^2 = 13 $$ 4. **Take the square root of both sides:** $$ x + 1 = \pm \sqrt{13} $$ 5. **Solve for $x$ by subtracting 1 from both sides:** $$ x = -1 \pm \sqrt{13} $$ **Final answer:** $$ x = -1 + \sqrt{13} \quad \text{or} \quad x = -1 - \sqrt{13} $$