Solve by factoring. r² + 12r + 32 = 0 The solution is r =

To solve the equation ( r² + 12r + 32 = 0 ) by factoring, we want to express the quadratic as a product of two binomials.

1. Start with the quadratic: ( r² + 12r + 32 ).
2. We need to find two numbers that multiply to ( 32 ) (the constant term) and add up to ( 12 ) (the coefficient of ( r )).
3. The numbers ( 4 ) and ( 8 ) satisfy this (since ( 4 times 8 = 32 ) and ( 4 + 8 = 12 )).
4. Rewrite the quadratic as: ( (r + 4)(r + 8) = 0 ).
5. Now, set each factor to zero:

– ( r + 4 = 0 ) → ( r = -4 )

– ( r + 8 = 0 ) → ( r = -8 )

So the solutions are ( r = -4, -8 ).