Simplify the Expression: Applying Distributive Property to 2(ab-7)(3+a)

To solve this problem, we'll follow these steps:

• Step 1: Distribute the expressions inside the parentheses.
• Step 2: Multiply and simplify expressions inside first, then outside.

Let's work through each step:

Step 1: Consider the expression $(ab-7)(3+a)$. Apply the distributive property:

• First, distribute $ab$ to both terms inside the second parentheses:
 $ab \cdot 3 = 3ab$
 $ab \cdot a = a^2b$

• Next, distribute $-7$ to both terms inside the second parentheses:
 $-7 \cdot 3 = -21$
 $-7 \cdot a = -7a$

Combining these, we have:

$(ab - 7)(3 + a) = 3ab + a^2b - 21 - 7a$.

Step 2: Multiply through by the factor $2$ outside the parentheses:

 $2 \cdot 3ab = 6ab$
 $2 \cdot a^2b = 2a^2b$
 $2 \cdot -21 = -42$
 $2 \cdot -7a = -14a$

Thus, our expression becomes:

$6ab + 2a^2b - 42 - 14a$.

Therefore, the solution to the problem is $6ab + 2a^2b - 42 - 14a$.