Question: C. Factoring Perfect Square Trinomials :\nUse the formula $a^2 + 2ab + b^2 = (a + b)^2$ or $a^2 - 2ab + b^2 = (a - b)^2$\n\n4. The area of a square courtyard is represented by $x^2 + 10x + 25$. Find the length of one side.\n\nD. Factoring Quadratic Trinomials :\n5. The area of a playground is $x^2 + 11x + 24$. Find the dimensions by factoring.

1. **Problem 4:** Find the length of one side of a square courtyard whose area is given by $x^2 + 10x + 25$.\n\n2. The formula for factoring perfect square trinomials is:\n$$a^2 + 2ab + b^2 = (a + b)^2$$\nor\n$$a^2 - 2ab + b^2 = (a - b)^2$$\n\n3. Identify $a^2$, $2ab$, and $b^2$ in the expression $x^2 + 10x + 25$.\n- Here, $a^2 = x^2$ so $a = x$.\n- $b^2 = 25$ so $b = 5$.\n- Check if $2ab = 10x$: $2 \times x \times 5 = 10x$, which matches.\n\n4. Since it fits the pattern $a^2 + 2ab + b^2$, factor as:\n$$x^2 + 10x + 25 = (x + 5)^2$$\n\n5. The length of one side of the square courtyard is therefore:\n$$x + 5$$\n\n---\n\n6. **Problem 5:** Find the dimensions of a playground with area $x^2 + 11x + 24$ by factoring.\n\n7. To factor a quadratic trinomial $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$.\n\n8. For $x^2 + 11x + 24$, find two numbers that multiply to $24$ and add to $11$.\n- These numbers are $8$ and $3$ because $8 \times 3 = 24$ and $8 + 3 = 11$.\n\n9. Factor the trinomial as:\n$$x^2 + 11x + 24 = (x + 8)(x + 3)$$\n\n10. The dimensions of the playground are therefore:\n$$x + 8$$ and $$x + 3$$\n