Imani and Todd were trying to solve the equation: \( x^{2}+6 x+5=8 \) Imani said, "I can factor the left-hand side into \( (x+1)(x+5) \), so l" solve using the zero product property." Todd said, "I can solve by completing the square. If I add 4 to each sid can rewrite the equation as \( (x+3)^{2}=12 \)." Whose solution strategy would work?

We start with the equation   x² + 6x + 5 = 8. The typical strategy is to bring all terms to one side. Subtract 8 from both sides to obtain:   x² + 6x + 5 – 8 = 0   x² + 6x – 3 = 0. Now, examine the two approaches: 1. Imani’s Approach:  • Imani noted that x² + 6x + 5 factors as (x + 1)(x + 5).  • However, if we factor the left-hand side in the original equation, we get:   (x + 1)(x + 5) = 8—not 0.  • The zero product property applies when a product equals 0. Since (x + 1)(x + 5) equals 8 (not 0), you cannot set each factor equal to 0 and solve.  • Thus, Imani’s plan to “solve using the zero product property” won’t work directly without first setting the equation equal to zero. 2. Todd’s Approach:  • Todd uses the method of completing the square.  • Starting from x² + 6x – 3 = 0 (or from the original form after subtracting the constant), he can rearrange:   x² + 6x = 3.  • To complete the square, take half of 6 (which is 3) and square it (3² = 9). Add 9 to both sides:   x² + 6x + 9 = 3 + 9,   (x + 3)² = 12.  • Now, taking square roots gives:   x + 3 = ±√12 = ±2√3,   and solving for x gives:   x = –3 ± 2√3.  • Todd’s method is correct (even if his comment mentioned “adding 4,” the proper number needed to complete the square is 9 in this case). Conclusion:  • Todd’s method of completing the square works (with the correct constant) and leads to the correct answers.  • Imani’s strategy of factoring does not directly work because factoring the left-hand side gives an expression that equals 8 instead of 0, making the zero product property inapplicable without further adjustments. Therefore, Todd’s solution strategy is the one that works.