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

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

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

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

Reorder the terms:
((xy + x2) + 3y * (x + y)) = 0
((xy + x2) + 3y * (x + y)) = 0
(xy + x2 + (x * 3y + y * 3y)) = 0
(xy + x2 + (3xy + 3y2)) = 0

Reorder the terms:
(xy + 3xy + x2 + 3y2) = 0

Combine like terms: xy + 3xy = 4xy
(4xy + x2 + 3y2) = 0

Solving
4xy + x2 + 3y2 = 0

Solving for variable 'x'.

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

Subproblem 1
Set the factor '(x + y)' equal to zero and attempt to solve:

Simplifying
x + y = 0

Solving
x + y = 0

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

Add '-1y' to each side of the equation.
x + y + -1y = 0 + -1y

Combine like terms: y + -1y = 0
x + 0 = 0 + -1y
x = 0 + -1y
Remove the zero:
x = -1y

Simplifying
x = -1y
Subproblem 2
Set the factor '(x + 3y)' equal to zero and attempt to solve:

Simplifying
x + 3y = 0

Solving
x + 3y = 0

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

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

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

Simplifying
x = -3y
Solution
x = {-1y, -3y}