1. **State the problem:** Solve the equation $$4x^2 + 7 = 71$$ for $$x$$. 2. **Isolate the term with $$x$$:** Subtract 7 from both sides: $$4x^2 + 7 - 7 = 71 - 7$$ $$4x^2 = 64$$ 3. **Divide both sides by 4 to solve for $$x^2$$:** $$\frac{4x^2}{\cancel{4}} = \frac{64}{\cancel{4}}$$ $$x^2 = 16$$ 4. **Take the square root of both sides:** $$x = \pm \sqrt{16}$$ 5. **Simplify the square root:** $$x = \pm 4$$ **Final answer:** $$x = \pm 4$$