Find the roots of the polynomial $\frac{x^2+x-2}{x^2+5x+6}$ by putting it in the form of an equation and then set it equal to zero

Factor the trinomial $x^2+5x+6$ finding two numbers that multiply to form $6$ and added form $5$

Rewrite the polynomial as the product of two binomials consisting of the sum of the variable and the found values

Factor the trinomial $x^2+x-2$ finding two numbers that multiply to form $-2$ and added form $1$

Rewrite the polynomial as the product of two binomials consisting of the sum of the variable and the found values

Simplifying

Multiply both sides of the equation by $x+3$

Any expression multiplied by $0$ is equal to $0$

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $-1$ from both sides of the equation

The valid solutions to the equation are the ones that, when replaced in the original equation, don't make any denominator equal to $0$, since division by zero is not allowed