=(2x+3y)(4x+5y)

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Solution for =(2x+3y)(4x+5y) equation: 


Simplifying
0 = (2x + 3y)(4x + 5y)

Multiply (2x + 3y) * (4x + 5y)
0 = (2x * (4x + 5y) + 3y * (4x + 5y))
0 = ((4x * 2x + 5y * 2x) + 3y * (4x + 5y))

Reorder the terms:
0 = ((10xy + 8x2) + 3y * (4x + 5y))
0 = ((10xy + 8x2) + 3y * (4x + 5y))
0 = (10xy + 8x2 + (4x * 3y + 5y * 3y))
0 = (10xy + 8x2 + (12xy + 15y2))

Reorder the terms:
0 = (10xy + 12xy + 8x2 + 15y2)

Combine like terms: 10xy + 12xy = 22xy
0 = (22xy + 8x2 + 15y2)

Solving
0 = 22xy + 8x2 + 15y2

Solving for variable 'x'.
Remove the zero:
-22xy + -8x2 + -15y2 = 22xy + 8x2 + 15y2 + -22xy + -8x2 + -15y2

Reorder the terms:
-22xy + -8x2 + -15y2 = 22xy + -22xy + 8x2 + -8x2 + 15y2 + -15y2

Combine like terms: 22xy + -22xy = 0
-22xy + -8x2 + -15y2 = 0 + 8x2 + -8x2 + 15y2 + -15y2
-22xy + -8x2 + -15y2 = 8x2 + -8x2 + 15y2 + -15y2

Combine like terms: 8x2 + -8x2 = 0
-22xy + -8x2 + -15y2 = 0 + 15y2 + -15y2
-22xy + -8x2 + -15y2 = 15y2 + -15y2

Combine like terms: 15y2 + -15y2 = 0
-22xy + -8x2 + -15y2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(22xy + 8x2 + 15y2) = 0

Factor a trinomial.
-1((4x + 5y)(2x + 3y)) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(4x + 5y)' equal to zero and attempt to solve:

Simplifying
4x + 5y = 0

Solving
4x + 5y = 0

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

Add '-5y' to each side of the equation.
4x + 5y + -5y = 0 + -5y

Combine like terms: 5y + -5y = 0
4x + 0 = 0 + -5y
4x = 0 + -5y
Remove the zero:
4x = -5y

Divide each side by '4'.
x = -1.25y

Simplifying
x = -1.25y

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

Simplifying
2x + 3y = 0

Solving
2x + 3y = 0

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

Add '-3y' to each side of the equation.
2x + 3y + -3y = 0 + -3y

Combine like terms: 3y + -3y = 0
2x + 0 = 0 + -3y
2x = 0 + -3y
Remove the zero:
2x = -3y

Divide each side by '2'.
x = -1.5y

Simplifying
x = -1.5y
Solution
x = {-1.25y, -1.5y}

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