Answer
$ \frac{a-9}{42a^2}$.
Work Step by Step
The given expression is
$\Rightarrow \frac{a^2-8a+15}{2a^3-10a^2}\cdot \frac{2a^2+3a}{3a^3-27a}\div \frac{14a+21}{a^2-6a-27}$
Invert the divisor and multiply.
$\Rightarrow \frac{a^2-8a+15}{2a^3-10a^2}\cdot \frac{2a^2+3a}{3a^3-27a}\cdot \frac{a^2-6a-27}{14a+21}$
Factor each numerator and denominator as shown below.
$\Rightarrow a^2-8a+15$
Rewrite the middle term $-8a$ as $-5a-3a$.
$\Rightarrow a^2-5a-3a+15$
Group terms.
$\Rightarrow (a^2-5a)+(-3a+15)$
Factor each group.
$\Rightarrow a(a-5)-3(a-5)$
Factor out $(a-5)$.
$\Rightarrow (a-5)(a-3)$
$\Rightarrow 2a^3-10a^2$
Factor out $2a^2$.
$\Rightarrow 2a^2(a-5)$
$\Rightarrow 2a^2+3a$
Factor out $a$.
$\Rightarrow a(2a+3)$
$\Rightarrow 3a^3-27a$
Factor out $3a$.
$\Rightarrow 3a(a^2-9)$
$\Rightarrow 3a(a^2-3^2)$
Use the special formula $(a^2-b^2)=(a+b)(a-b)$
$\Rightarrow 3a(a+3)(a-3)$
$\Rightarrow a^2-6a-27$
Rewrite the middle term $-6a$ as $-9a+3a$.
$\Rightarrow a^2-9a+3a-27$
Group terms.
$\Rightarrow (a^2-9a)+(3a-27)$
Factor each group.
$\Rightarrow a(a-9)+3(a-9)$
Factor out $(a-9)$.
$\Rightarrow (a-9)(a+3)$
$\Rightarrow 14a+21$
Factor out $7$.
$\Rightarrow 7(2a+3)$
Substitute all the factors into the given expression.
$\Rightarrow \frac{(a-5)(a-3)}{2a^2(a-5)}\cdot \frac{a(2a+3)}{3a(a+3)(a-3)}\cdot \frac{(a-9)(a+3)}{7(2a+3)}$
Cancel common terms.
$\Rightarrow \frac{a-9}{42a^2}$.
