1. **State the problem:** Solve the equation $$3(2x+1)^2 = 2(52+1)$$ for $x$. 2. **Simplify the right side:** Calculate $52+1$. $$52+1=53$$ So the equation becomes: $$3(2x+1)^2 = 2 \times 53$$ 3. **Multiply the right side:** $$2 \times 53 = 106$$ So the equation is: $$3(2x+1)^2 = 106$$ 4. **Isolate the squared term:** Divide both sides by 3. $$\frac{3(2x+1)^2}{\cancel{3}} = \frac{106}{\cancel{3}}$$ $$ (2x+1)^2 = \frac{106}{3} $$ 5. **Take the square root of both sides:** $$ 2x+1 = \pm \sqrt{\frac{106}{3}} $$ 6. **Isolate $x$:** Subtract 1 from both sides. $$ 2x = -1 \pm \sqrt{\frac{106}{3}} $$ 7. **Divide both sides by 2:** $$ x = \frac{-1 \pm \sqrt{\frac{106}{3}}}{2} $$ **Final answer:** $$ x = \frac{-1 + \sqrt{\frac{106}{3}}}{2} \quad \text{or} \quad x = \frac{-1 - \sqrt{\frac{106}{3}}}{2} $$