(x+4)2-(x+3)(x-3)=1

Solution for (x+4)2-(x+3)(x-3)=1 equation:

Simplifying
(x + 4) * 2 + -1(x + 3)(x + -3) = 1

Reorder the terms:
(4 + x) * 2 + -1(x + 3)(x + -3) = 1

Reorder the terms for easier multiplication:
2(4 + x) + -1(x + 3)(x + -3) = 1
(4 * 2 + x * 2) + -1(x + 3)(x + -3) = 1
(8 + 2x) + -1(x + 3)(x + -3) = 1

Reorder the terms:
8 + 2x + -1(3 + x)(x + -3) = 1

Reorder the terms:
8 + 2x + -1(3 + x)(-3 + x) = 1

Multiply (3 + x) * (-3 + x)
8 + 2x + -1(3(-3 + x) + x(-3 + x)) = 1
8 + 2x + -1((-3 * 3 + x * 3) + x(-3 + x)) = 1
8 + 2x + -1((-9 + 3x) + x(-3 + x)) = 1
8 + 2x + -1(-9 + 3x + (-3 * x + x * x)) = 1
8 + 2x + -1(-9 + 3x + (-3x + x2)) = 1

Combine like terms: 3x + -3x = 0
8 + 2x + -1(-9 + 0 + x2) = 1
8 + 2x + -1(-9 + x2) = 1
8 + 2x + (-9 * -1 + x2 * -1) = 1
8 + 2x + (9 + -1x2) = 1

Reorder the terms:
8 + 9 + 2x + -1x2 = 1

Combine like terms: 8 + 9 = 17
17 + 2x + -1x2 = 1

Solving
17 + 2x + -1x2 = 1

Solving for variable 'x'.

Reorder the terms:
17 + -1 + 2x + -1x2 = 1 + -1

Combine like terms: 17 + -1 = 16
16 + 2x + -1x2 = 1 + -1

Combine like terms: 1 + -1 = 0
16 + 2x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-16 + -2x + x2 = 0

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + -2x + 16 + x2 = 0 + 16

Reorder the terms:
-16 + 16 + -2x + x2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + -2x + x2 = 0 + 16
-2x + x2 = 0 + 16

Combine like terms: 0 + 16 = 16
-2x + x2 = 16

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = 16 + 1

Reorder the terms:
1 + -2x + x2 = 16 + 1

Combine like terms: 16 + 1 = 17
1 + -2x + x2 = 17

Factor a perfect square on the left side:
(x + -1)(x + -1) = 17

Calculate the square root of the right side: 4.123105626

Break this problem into two subproblems by setting 
(x + -1) equal to 4.123105626 and -4.123105626.

Subproblem 1
x + -1 = 4.123105626

Simplifying
x + -1 = 4.123105626

Reorder the terms:
-1 + x = 4.123105626

Solving
-1 + x = 4.123105626

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = 4.123105626 + 1

Combine like terms: -1 + 1 = 0
0 + x = 4.123105626 + 1
x = 4.123105626 + 1

Combine like terms: 4.123105626 + 1 = 5.123105626
x = 5.123105626

Simplifying
x = 5.123105626

Subproblem 2
x + -1 = -4.123105626

Simplifying
x + -1 = -4.123105626

Reorder the terms:
-1 + x = -4.123105626

Solving
-1 + x = -4.123105626

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = -4.123105626 + 1

Combine like terms: -1 + 1 = 0
0 + x = -4.123105626 + 1
x = -4.123105626 + 1

Combine like terms: -4.123105626 + 1 = -3.123105626
x = -3.123105626

Simplifying
x = -3.123105626

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.123105626, -3.123105626}