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?

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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.
Todd's method of completing the square works correctly to solve the equation.

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