Answer
$\dfrac{x - 8}{x - 10}$
Restrictions: $x \ne -3, 10$
Work Step by Step
Let's first take a look at the problem to see what we can cancel out from the numerator and denominator. We can factor out the polynomials to make this easier:
Let's look at the numerator first:
$x^2 - 5x - 24$
We can use $-8$ and $3$:
$(x - 8)(x + 3)$
Let's look at the polynomial in the denominator:
$x^2 - 7x - 30$
The factors $-10$ and $3$ will work:
$(x - 10)(x + 3)$
Let's put the factors back into the rational expression:
$\dfrac{(x - 8)(x + 3)}{(x - 10)(x + 3)}$
Factor out common factor $x + 3$:
$\frac{x - 8}{x - 10}$
To check what restrictions we have for the variables, we need to find which values of the variables will make the denominators of the original expressions equal $0$, which would make the fraction undefined. Let's set the denominator equal to $0$, and then solve:
$(x - 10)(x + 3) = 0$
Set each factor equal to zero, according to the zero product property:
First factor:
$x - 10 = 0$
Add $10$ to each side:
$x = 10$
Second factor:
$x + 3 = 0$
Subtract each side by $3$:
$x = -3$
Restriction: $x \ne -3, 10$
