Solve the equation x^2 + 3x = -2.

Step-by-step solution

1. 1

   Get a zero on one side. Factoring and the zero-product property both require standard form, so add $2$ to both sides:

   $x^2+3x+2=0$

2. 2

   Find the factor pair. With a leading coefficient of $1$, look for two numbers whose product is the constant $2$ and whose sum is the middle coefficient $3$. Those are $1$ and $2$:

   $x^2+3x+2=(x+1)(x+2)$

3. 3

   Set each factor to zero. A product is zero only if one of its factors is zero:

   $x+1=0\ \Rightarrow\ x=-1,\qquad x+2=0\ \Rightarrow\ x=-2$

4. 4

   Predict the signs as a sanity check. Both the sum ($-3$) and the product ($+2$) of the roots are consistent with two negative roots, which is what we found — a quick structural check before substituting.

5. 5

   Substitute into the original equation. At $x=-1$: $1-3=-2$ $\checkmark$. At $x=-2$: $4-6=-2$ $\checkmark$.