(6x+3)(10-x)=61-(x+1)

Solution for (6x+3)(10-x)=61-(x+1) equation:

Simplifying
(6x + 3)(10 + -1x) = 61 + -1(x + 1)

Reorder the terms:
(3 + 6x)(10 + -1x) = 61 + -1(x + 1)

Multiply (3 + 6x) * (10 + -1x)
(3(10 + -1x) + 6x * (10 + -1x)) = 61 + -1(x + 1)
((10 * 3 + -1x * 3) + 6x * (10 + -1x)) = 61 + -1(x + 1)
((30 + -3x) + 6x * (10 + -1x)) = 61 + -1(x + 1)
(30 + -3x + (10 * 6x + -1x * 6x)) = 61 + -1(x + 1)
(30 + -3x + (60x + -6x2)) = 61 + -1(x + 1)

Combine like terms: -3x + 60x = 57x
(30 + 57x + -6x2) = 61 + -1(x + 1)

Reorder the terms:
30 + 57x + -6x2 = 61 + -1(1 + x)
30 + 57x + -6x2 = 61 + (1 * -1 + x * -1)
30 + 57x + -6x2 = 61 + (-1 + -1x)

Combine like terms: 61 + -1 = 60
30 + 57x + -6x2 = 60 + -1x

Solving
30 + 57x + -6x2 = 60 + -1x

Solving for variable 'x'.

Reorder the terms:
30 + -60 + 57x + x + -6x2 = 60 + -1x + -60 + x

Combine like terms: 30 + -60 = -30
-30 + 57x + x + -6x2 = 60 + -1x + -60 + x

Combine like terms: 57x + x = 58x
-30 + 58x + -6x2 = 60 + -1x + -60 + x

Reorder the terms:
-30 + 58x + -6x2 = 60 + -60 + -1x + x

Combine like terms: 60 + -60 = 0
-30 + 58x + -6x2 = 0 + -1x + x
-30 + 58x + -6x2 = -1x + x

Combine like terms: -1x + x = 0
-30 + 58x + -6x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-15 + 29x + -3x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-15 + 29x + -3x2)' equal to zero and attempt to solve:

Simplifying
-15 + 29x + -3x2 = 0

Solving
-15 + 29x + -3x2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
5 + -9.666666667x + x2 = 0

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + -9.666666667x + -5 + x2 = 0 + -5

Reorder the terms:
5 + -5 + -9.666666667x + x2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + -9.666666667x + x2 = 0 + -5
-9.666666667x + x2 = 0 + -5

Combine like terms: 0 + -5 = -5
-9.666666667x + x2 = -5

The x term is -9.666666667x.  Take half its coefficient (-4.833333334).
Square it (23.36111112) and add it to both sides.

Add '23.36111112' to each side of the equation.
-9.666666667x + 23.36111112 + x2 = -5 + 23.36111112

Reorder the terms:
23.36111112 + -9.666666667x + x2 = -5 + 23.36111112

Combine like terms: -5 + 23.36111112 = 18.36111112
23.36111112 + -9.666666667x + x2 = 18.36111112

Factor a perfect square on the left side:
(x + -4.833333334)(x + -4.833333334) = 18.36111112

Calculate the square root of the right side: 4.284986712

Break this problem into two subproblems by setting 
(x + -4.833333334) equal to 4.284986712 and -4.284986712.

Subproblem 1
x + -4.833333334 = 4.284986712

Simplifying
x + -4.833333334 = 4.284986712

Reorder the terms:
-4.833333334 + x = 4.284986712

Solving
-4.833333334 + x = 4.284986712

Solving for variable 'x'.

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

Add '4.833333334' to each side of the equation.
-4.833333334 + 4.833333334 + x = 4.284986712 + 4.833333334

Combine like terms: -4.833333334 + 4.833333334 = 0.000000000
0.000000000 + x = 4.284986712 + 4.833333334
x = 4.284986712 + 4.833333334

Combine like terms: 4.284986712 + 4.833333334 = 9.118320046
x = 9.118320046

Simplifying
x = 9.118320046

Subproblem 2
x + -4.833333334 = -4.284986712

Simplifying
x + -4.833333334 = -4.284986712

Reorder the terms:
-4.833333334 + x = -4.284986712

Solving
-4.833333334 + x = -4.284986712

Solving for variable 'x'.

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

Add '4.833333334' to each side of the equation.
-4.833333334 + 4.833333334 + x = -4.284986712 + 4.833333334

Combine like terms: -4.833333334 + 4.833333334 = 0.000000000
0.000000000 + x = -4.284986712 + 4.833333334
x = -4.284986712 + 4.833333334

Combine like terms: -4.284986712 + 4.833333334 = 0.548346622
x = 0.548346622

Simplifying
x = 0.548346622

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {9.118320046, 0.548346622}
Solution
x = {9.118320046, 0.548346622}