Find the Missing Factor in (x+2)(□-4) = a+2a/x-4x-8

Complete the missing element

$(x+2)(?-4)=a+2\frac{a}{x}-4x-8$

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

• Step 1: Expand the given expression using distributive property.
• Step 2: Match each term to the provided expression.
• Step 3: Solve for the missing element.

Firstly, we need to expand the left side of the equation $(x+2)(k-4)$:

Applying the distributive property:
$(x+2)(k-4) = x(k-4) + 2(k-4)$.
Continue expanding:
$= xk - 4x + 2k - 8$.

Now, compare the simplified left hand expression $xk - 4x + 2k - 8$ with the right side of the given equation $a + 2\frac{a}{x} - 4x - 8$.

By matching terms:

• Coefficients of $-4x$ and $-8$ are already matching.
• The term $xk + 2k$ must equal $a + 2\frac{a}{x}$.

To create the term $2\frac{a}{x}$, we deduce that the missing value $k$ for $xk$ must be $\frac{a}{x}$, because substituting $\frac{a}{x}$ results in terms becoming $a$ and $2\frac{a}{x}$.

Therefore, the solution for the missing element is $\frac{a}{x}$.