Question

which expression is equal to \\(\frac{6a^2 - 30a}{a - 2} \cdot \frac{a^2 + 2a - 8}{2a^3 - 10a^2}\\) ? \\(\frac{3a + 12}{a}\\) \\(\frac{3(a - 8)(a + 4)}{2a(a + 5)}\\) \\(\frac{2a(2a - 8)}{a^4 - 10}\\) \\(\frac{3a + 4}{a}\\)

Explanation:

Step1: Factor all numerators/denominators

1. $6a^2 - 30a = 6a(a - 5)$
2. $a^2 + 2a - 8 = (a + 4)(a - 2)$
3. $2a^3 - 10a^2 = 2a^2(a - 5)$

Step2: Rewrite the product

$$\frac{6a(a - 5)}{a - 2} \cdot \frac{(a + 4)(a - 2)}{2a^2(a - 5)}$$

Step3: Cancel common factors

Cancel $a-5$, $a-2$, $2a$ from numerator/denominator:
$$\frac{6a \cdot (a+4)}{2a^2} = \frac{6(a+4)}{2a}$$

Step4: Simplify the remaining terms

$$\frac{6(a+4)}{2a} = \frac{3(a+4)}{a} = \frac{3a + 12}{a}$$

Answer:

$\frac{3a+12}{a}$