Factor the Expression: 21ab - 63a²/b - 14ba²

To solve the problem of decomposing the expression $21ab - \frac{63a^2}{b} - 14ba^2$ into factors, we will follow these steps:

• Step 1: Identify the Greatest Common Factor (GCF).
• Step 2: Factor out the GCF.
• Step 3: Verify the result by expanding to check that it matches the original expression.

Let's go through each step:

Step 1: Find the GCF of the terms. Examine each term in the expression:

- The three terms are $21ab$, $-\frac{63a^2}{b}$, and $-14ba^2$.
- The coefficients $21$, $\frac{63}{1}$, and $14$ share a common factor of $7$.
- The variables $a$ and $b$ are present in each term. Each term has at least one $a$ and one $b$.
- Thus, the GCF is $7ab$.

Step 2: Factor out the GCF $7ab$ from the expression:

$21ab = 7ab \times 3$
$-\frac{63a^2}{b} = 7ab \times \left(-\frac{9a}{b^2}\right)$
$-14ba^2 = 7ab \times (-2a)$

Combining these, the expression factors as:

$7ab(3 - \frac{9a}{b^2} - 2a)$

Step 3: Verify by expanding the factored expression:

- Expanding $7ab(3 - \frac{9a}{b^2} - 2a)$:
$7ab \times 3 = 21ab$
$7ab \times \left(-\frac{9a}{b^2}\right) = -\frac{63a^2}{b}$
$7ab \times (-2a) = -14ba^2$

These match the original expression, confirming the factorization is correct.

Therefore, the solution to the problem is $7ab(3-\frac{9a}{b^2}-2a)$.