Answer
$\frac{y + 6}{y - 2}$
Restriction: $y \ne 2$
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:
$2y^2 + 8y - 24$
We need to look at which factors when multiplied together equal the first coefficient multiplied with the constant ($(2)(-24)$ or $-48$), but when added together will equal the middle coefficient $8$:
It looks like $12$ and $-4$ will work.
Let's use them to split the middle term:
$2y^2 + 12y - 4y - 24$
Group the first two terms and the second two terms:
$(2y^2 + 12y) + (- 4y - 24)$
Factor out common factors:
$2y(y + 6) - 4(y + 6)$
Now, let's keep the common binomial factor and combine the coefficients into another factor:
$(2y - 4)(y + 6)$
Let's look at the polynomial in the denominator:
$2y^2 - 8y + 8$
We need to look at which factors when multiplied together equal the first coefficient multiplied with the constant ($(2)(8)$ or $16$), but when added together will equal the middle coefficient $-8$:
It looks like $-4$ and $-4$ will work.
Let's use them to split the middle term:
$2y^2 - 4y - 4y + 8$
Group the first two terms and the second two terms:
$(2y^2 - 4y) + (- 4y + 8)$
Factor out common factors:
$2y(y - 2) - 4(y - 2)$
Now, let's keep the common binomial factor and combine the coefficients into another factor:
$(2y - 4)(y - 2)$
Let's put the factors back into the rational expression:
$\frac{(2y - 4)(y + 6)}{(2y - 4)(y - 2)}$
Factor out common factor $2y - 4$:
$\frac{y + 6}{y - 2}$
To check what restrictions we have for the variable, we need to find which values of the variable will make the denominator of the original expression equal $0$, which would make the fraction undefined. Let's set the denominator equal to $0$, and then solve:
$(2y - 4)(y - 2) = 0$
Set each factor equal to zero, according to the zero product property:
First factor:
$2y - 4 = 0$
Add $4$ to each side:
$2y = 4$
Divide each side by $2$:
$y = 2$
Second factor:
$y - 2 = 0$
Add $2$ to each side:
$y = 2$
Restriction: $y \ne 2$
