actorize; $$ (2 x+3 y)^{2}-(x-4 y)^{2} $$

Question

UpStudy AI · Accepted Answer

We start with the expression

$$
(2x+3y)^2 - (x-4y)^2.
$$

This is a difference of squares, which factors as

$$
a^2 - b^2 = (a-b)(a+b).
$$

Let

$$
a = 2x+3y \quad \text{and} \quad b = x-4y.
$$

Applying the formula, we have

$$
(2x+3y)^2 - (x-4y)^2 = \big[(2x+3y) - (x-4y)\big] \cdot \big[(2x+3y) + (x-4y)\big].
$$

Now, simplify each factor:

1. For the first factor:
   $$
   (2x+3y) - (x-4y) = 2x+3y - x + 4y = (2x-x) + (3y+4y) = x+7y.
   $$

2. For the second factor:
   $$
   (2x+3y) + (x-4y) = 2x+3y+x-4y = (2x+x) + (3y-4y) = 3x-y.
   $$

Thus, the factorized form is

$$
(x+7y)(3x-y).
$$
The expression $$ (2x+3y)^2 - (x-4y)^2 $$ factors to $$ (x + 7y)(3x - y) $$.

actorize;