.5(3x+15)(x+2)=105

Solution for .5(3x+15)(x+2)=105 equation:

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

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

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

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

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

Solving
15 + 10.5x + 1.5x2 = 105

Solving for variable 'x'.

Reorder the terms:
15 + -105 + 10.5x + 1.5x2 = 105 + -105

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

Combine like terms: 105 + -105 = 0
-90 + 10.5x + 1.5x2 = 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}