Solve (x-4y)(2x+?): Find the Missing Term in Polynomial Expansion

Fill in the missing number

$(x-4y)(2x+?)=2x^2-12y-8xy+3$

To solve the problem, we will expand $(x-4y)(2x+?)$ using the distributive property and match it to the given polynomial:

First, expand the expression:
$(x-4y)(2x+?) = x(2x+?) - 4y(2x+?)$

Upon expanding, we get:
$= x \cdot 2x + x \cdot ? - 4y \cdot 2x - 4y \cdot ?$ $= 2x^2 + x \cdot ? - 8xy - 4y \times ?$

We equate the expanded expression to the given polynomial $2x^2 - 8xy - 12y + 3$:
$2x^2 + x \times ? - 8xy - 4y \times ? = 2x^2 - 8xy - 12y + 3$

By matching terms, we see:
1. The $x \cdot ?$ + $-4y \cdot ?$ needs to compensate for $-12y$ and the constant 3.
2. Equate negative constant and remaining components:
$-4y \times ? = -12y$ Therefore, $? = \frac{-12y + 3}{-4y} = 3$.

After calculation, the missing number aligns with the given polynomial. Therefore, the missing number is:

$3$.