1. **State the problem:** We are given a rectangle with sides AG and NL. The length of side AG is $x^2$ and the length of side NL is $3x - 4$. Since opposite sides of a rectangle are equal, we have $AG = NL$. 2. **Set up the equation:** Because AG and NL are opposite sides of the rectangle, they must be equal: $$x^2 = 3x - 4$$ 3. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 - 3x + 4 = 0$$ 4. **Solve the quadratic equation:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-3$, and $c=4$. Calculate the discriminant: $$\Delta = b^2 - 4ac = (-3)^2 - 4(1)(4) = 9 - 16 = -7$$ 5. **Interpret the discriminant:** Since $\Delta < 0$, there are no real solutions for $x$. This means the given lengths cannot form a rectangle with real side lengths. **Final answer:** There is no real value of $x$ that satisfies $x^2 = 3x - 4$ for the rectangle sides.