Factorize the Expression: 16xa² + 80x/a - 40a³

Decompose the following expression into factors:

$16xa^2+\frac{80x}{a}-40a^3$

To solve the expression $16xa^2 + \frac{80x}{a} - 40a^3$ by decomposing it into factors, we need to follow a series of detailed steps:

Step 1: Identify common factors among the terms.
Observe that the terms are $16xa^2$, $\frac{80x}{a}$, and $-40a^3$.
- Each term involves either $x$ or $a$. The coefficient common factor is 8.

Step 2: Extract the common factor.
Examine each term for their factors.
- $16xa^2 = 8 \cdot 2xa^2$
- $\frac{80x}{a} = 8 \cdot \frac{10x}{a}$
- $-40a^3 = 8 \cdot (-5a^3)$
The common factor across all three is $8$.

Step 3: Factor out the common factor.
We factor out the common 8, resulting in:
$8(2xa^2 + \frac{10x}{a} - 5a^3)$.

Step 4: Further observe terms to simplify.
Notice that within $8(...)$, each term can factor out one more common factor, which is $a$:
- $2xa^2 = a \cdot 2ax$
- $\frac{10x}{a} = a \cdot \frac{10x}{a^2}$
- $-5a^3 = a \cdot (-5a^2)$
Thus, we factor out $a$, yielding:

Thus, the expression simplifies as:
$8a(2ax + \frac{10x}{a^2} - 5a^2)$.

Therefore, the decomposed form of the expression is $8a(2ax+\frac{10x}{a^2}-5a^2)$, matching choice 1.