y^2+5zy+4z^2=0

Solution for y^2+5zy+4z^2=0 equation:

Simplifying
y2 + 5zy + 4z2 = 0

Reorder the terms:
5yz + y2 + 4z2 = 0

Solving
5yz + y2 + 4z2 = 0

Solving for variable 'y'.

Factor a trinomial.
(y + z)(y + 4z) = 0

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

Simplifying
y + z = 0

Solving
y + z = 0

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

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

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

Simplifying
y = -1z
Subproblem 2
Set the factor '(y + 4z)' equal to zero and attempt to solve:

Simplifying
y + 4z = 0

Solving
y + 4z = 0

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

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

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

Simplifying
y = -4z
Solution
y = {-1z, -4z}