Answer
$-\frac{2a+1}{a+3}$
$a\ne\frac{5}{2}$ and $a \ne -3$
Work Step by Step
To simplify the expression, we need to arrange the numerator and denominator in descending powers and then factor the numerator and denominator.
$\frac{4a^2 - 8a - 5}{15 - a - 2a^2}$ = $\frac{4a^2 - 8a - 5}{-2a^2 - a +15}$
Now we need to factor out a negative from the denominator and then factor:
$-\frac{4a^2 - 8a - 5}{2a^2 + a -15}$ = $-\frac{(2a+1)(2a-5)}{(2a-5)(a+3)}$
Now we can divide out the common factor of $(2a-5)$
$-\frac{(2a+1)(2a-5)}{(2a-5)(a+3)} = -\frac{2a+1}{a+3}$
To find the excluded values, we need to look at the factored expression before simplifying:
$-\frac{(2a+1)(2a-5)}{(2a-5)(a+3)}$
Setting each factor of the denominator equal to 0, we will find the excluded values:
$2a - 5 = 0$
$2a=5$
$a=\frac{5}{2}$
$a+3 = 0$
$a = -3$
Therefore, $a\ne\frac{5}{2}$ and $a \ne -3$
