Final Answer
Step-by-step Solution
Specify the solving method
Find the roots of the polynomial $\frac{\frac{x^2-16}{x-1}}{\frac{x^2+6x+8}{x^2+2x-3}}$ by putting it in the form of an equation and then set it equal to zero
Simplify the fraction $\frac{\frac{x^2-16}{x-1}}{\frac{x^2+6x+8}{x^2+2x-3}}$
Factor the trinomial $\left(x^2+2x-3\right)$ finding two numbers that multiply to form $-3$ and added form $2$
Thus
Simplifying
Factor the trinomial $x^2+6x+8$ finding two numbers that multiply to form $8$ and added form $6$
Thus
Multiply both sides of the equation by $\left(x+2\right)\left(x+4\right)$
Break the equation in $2$ factors and set each equal to zero, to obtain
Solve the equation ($1$)
We need to isolate the dependent variable , we can do that by simultaneously subtracting $-16$ from both sides of the equation
Canceling terms on both sides
Removing the variable's exponent
Cancel exponents $2$ and $\frac{1}{2}$
The square root of $16$ is
As in the equation we have the sign $\pm$, this produces two identical equations that differ in the sign of the term $4$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign
Solve the equation ($2$)
We need to isolate the dependent variable , we can do that by simultaneously subtracting $3$ from both sides of the equation
Canceling terms on both sides
Combining all solutions, the $3$ solutions of the equation are
Verify that the solutions obtained are valid in the initial 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