105=0.5(x+2)(3x+15)

Solution for 105=0.5(x+2)(3x+15) equation:

Simplifying
105 = 0.5(x + 2)(3x + 15)

Reorder the terms:
105 = 0.5(2 + x)(3x + 15)

Reorder the terms:
105 = 0.5(2 + x)(15 + 3x)

Multiply (2 + x) * (15 + 3x)
105 = 0.5(2(15 + 3x) + x(15 + 3x))
105 = 0.5((15 * 2 + 3x * 2) + x(15 + 3x))
105 = 0.5((30 + 6x) + x(15 + 3x))
105 = 0.5(30 + 6x + (15 * x + 3x * x))
105 = 0.5(30 + 6x + (15x + 3x2))

Combine like terms: 6x + 15x = 21x
105 = 0.5(30 + 21x + 3x2)
105 = (30 * 0.5 + 21x * 0.5 + 3x2 * 0.5)
105 = (15 + 10.5x + 1.5x2)

Solving
105 = 15 + 10.5x + 1.5x2

Solving for variable 'x'.

Combine like terms: 105 + -15 = 90
90 + -10.5x + -1.5x2 = 15 + 10.5x + 1.5x2 + -15 + -10.5x + -1.5x2

Reorder the terms:
90 + -10.5x + -1.5x2 = 15 + -15 + 10.5x + -10.5x + 1.5x2 + -1.5x2

Combine like terms: 15 + -15 = 0
90 + -10.5x + -1.5x2 = 0 + 10.5x + -10.5x + 1.5x2 + -1.5x2
90 + -10.5x + -1.5x2 = 10.5x + -10.5x + 1.5x2 + -1.5x2

Combine like terms: 10.5x + -10.5x = 0.0
90 + -10.5x + -1.5x2 = 0.0 + 1.5x2 + -1.5x2
90 + -10.5x + -1.5x2 = 1.5x2 + -1.5x2

Combine like terms: 1.5x2 + -1.5x2 = 0.0
90 + -10.5x + -1.5x2 = 0.0

Begin completing the square.  Divide all terms by
-1.5 the coefficient of the squared term: 

Divide each side by '-1.5'.
-60 + 7x + x2 = 0

Move the constant term to the right:

Add '60' to each side of the equation.
-60 + 7x + 60 + x2 = 0 + 60

Reorder the terms:
-60 + 60 + 7x + x2 = 0 + 60

Combine like terms: -60 + 60 = 0
0 + 7x + x2 = 0 + 60
7x + x2 = 0 + 60

Combine like terms: 0 + 60 = 60
7x + x2 = 60

The x term is 7x.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7x + 12.25 + x2 = 60 + 12.25

Reorder the terms:
12.25 + 7x + x2 = 60 + 12.25

Combine like terms: 60 + 12.25 = 72.25
12.25 + 7x + x2 = 72.25

Factor a perfect square on the left side:
(x + 3.5)(x + 3.5) = 72.25

Calculate the square root of the right side: 8.5

Break this problem into two subproblems by setting 
(x + 3.5) equal to 8.5 and -8.5.

Subproblem 1
x + 3.5 = 8.5

Simplifying
x + 3.5 = 8.5

Reorder the terms:
3.5 + x = 8.5

Solving
3.5 + x = 8.5

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = 8.5 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = 8.5 + -3.5
x = 8.5 + -3.5

Combine like terms: 8.5 + -3.5 = 5
x = 5

Simplifying
x = 5

Subproblem 2
x + 3.5 = -8.5

Simplifying
x + 3.5 = -8.5

Reorder the terms:
3.5 + x = -8.5

Solving
3.5 + x = -8.5

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = -8.5 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = -8.5 + -3.5
x = -8.5 + -3.5

Combine like terms: -8.5 + -3.5 = -12
x = -12

Simplifying
x = -12

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5, -12}