(2x+4y)(3x+4y)=0

Solution for (2x+4y)(3x+4y)=0 equation:

Simplifying
(2x + 4y)(3x + 4y) = 0

Multiply (2x + 4y) * (3x + 4y)
(2x * (3x + 4y) + 4y * (3x + 4y)) = 0
((3x * 2x + 4y * 2x) + 4y * (3x + 4y)) = 0

Reorder the terms:
((8xy + 6x2) + 4y * (3x + 4y)) = 0
((8xy + 6x2) + 4y * (3x + 4y)) = 0
(8xy + 6x2 + (3x * 4y + 4y * 4y)) = 0
(8xy + 6x2 + (12xy + 16y2)) = 0

Reorder the terms:
(8xy + 12xy + 6x2 + 16y2) = 0

Combine like terms: 8xy + 12xy = 20xy
(20xy + 6x2 + 16y2) = 0

Solving
20xy + 6x2 + 16y2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(10xy + 3x2 + 8y2) = 0

Factor a trinomial.
2((3x + 4y)(x + 2y)) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(3x + 4y)' equal to zero and attempt to solve:

Simplifying
3x + 4y = 0

Solving
3x + 4y = 0

Move all terms containing x to the left, all other terms to the right.

Add '-4y' to each side of the equation.
3x + 4y + -4y = 0 + -4y

Combine like terms: 4y + -4y = 0
3x + 0 = 0 + -4y
3x = 0 + -4y
Remove the zero:
3x = -4y

Divide each side by '3'.
x = -1.333333333y

Simplifying
x = -1.333333333y
Subproblem 2
Set the factor '(x + 2y)' equal to zero and attempt to solve:

Simplifying
x + 2y = 0

Solving
x + 2y = 0

Move all terms containing x to the left, all other terms to the right.

Add '-2y' to each side of the equation.
x + 2y + -2y = 0 + -2y

Combine like terms: 2y + -2y = 0
x + 0 = 0 + -2y
x = 0 + -2y
Remove the zero:
x = -2y

Simplifying
x = -2y
Solution
x = {-1.333333333y, -2y}