1. **State the problem:** We have a rectangle with area given by $A = x^2 + 5x$ square meters. 2. **Part (a): Find the dimensions if the area is 24 m².** We set the area equal to 24: $$x^2 + 5x = 24$$ 3. **Rewrite the equation:** $$x^2 + 5x - 24 = 0$$ 4. **Use the quadratic formula:** For $ax^2 + bx + c = 0$, solutions are $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=1$, $b=5$, $c=-24$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 5^2 - 4(1)(-24) = 25 + 96 = 121$$ 6. **Find the roots:** $$x = \frac{-5 \pm \sqrt{121}}{2} = \frac{-5 \pm 11}{2}$$ 7. **Calculate each solution:** - $x = \frac{-5 + 11}{2} = \frac{6}{2} = 3$ - $x = \frac{-5 - 11}{2} = \frac{-16}{2} = -8$ 8. **Interpret the solutions:** Since $x$ represents a length, it must be positive, so $x=3$ is valid. 9. **Find the other dimension:** Area $= x \times (x+5)$, so the other side is $x+5$. For $x=3$, other side is $3 + 5 = 8$ meters. **Answer for (a):** Dimensions are 3 m and 8 m. 10. **Part (b): Discuss if two solutions exist and what they represent.** Two solutions exist mathematically: $x=3$ and $x=-8$. However, since length cannot be negative, $x=-8$ is not physically meaningful. Thus, only one valid dimension set exists. **Summary:** - Two solutions from quadratic equation. - Only positive solution represents a real rectangle dimension. - Negative solution is extraneous in this context.