(x+9)(3x+4)=(x+16)(x+3)

Solution for (x+9)(3x+4)=(x+16)(x+3) equation:

Simplifying
(x + 9)(3x + 4) = (x + 16)(x + 3)

Reorder the terms:
(9 + x)(3x + 4) = (x + 16)(x + 3)

Reorder the terms:
(9 + x)(4 + 3x) = (x + 16)(x + 3)

Multiply (9 + x) * (4 + 3x)
(9(4 + 3x) + x(4 + 3x)) = (x + 16)(x + 3)
((4 * 9 + 3x * 9) + x(4 + 3x)) = (x + 16)(x + 3)
((36 + 27x) + x(4 + 3x)) = (x + 16)(x + 3)
(36 + 27x + (4 * x + 3x * x)) = (x + 16)(x + 3)
(36 + 27x + (4x + 3x2)) = (x + 16)(x + 3)

Combine like terms: 27x + 4x = 31x
(36 + 31x + 3x2) = (x + 16)(x + 3)

Reorder the terms:
36 + 31x + 3x2 = (16 + x)(x + 3)

Reorder the terms:
36 + 31x + 3x2 = (16 + x)(3 + x)

Multiply (16 + x) * (3 + x)
36 + 31x + 3x2 = (16(3 + x) + x(3 + x))
36 + 31x + 3x2 = ((3 * 16 + x * 16) + x(3 + x))
36 + 31x + 3x2 = ((48 + 16x) + x(3 + x))
36 + 31x + 3x2 = (48 + 16x + (3 * x + x * x))
36 + 31x + 3x2 = (48 + 16x + (3x + x2))

Combine like terms: 16x + 3x = 19x
36 + 31x + 3x2 = (48 + 19x + x2)

Solving
36 + 31x + 3x2 = 48 + 19x + x2

Solving for variable 'x'.

Reorder the terms:
36 + -48 + 31x + -19x + 3x2 + -1x2 = 48 + 19x + x2 + -48 + -19x + -1x2

Combine like terms: 36 + -48 = -12
-12 + 31x + -19x + 3x2 + -1x2 = 48 + 19x + x2 + -48 + -19x + -1x2

Combine like terms: 31x + -19x = 12x
-12 + 12x + 3x2 + -1x2 = 48 + 19x + x2 + -48 + -19x + -1x2

Combine like terms: 3x2 + -1x2 = 2x2
-12 + 12x + 2x2 = 48 + 19x + x2 + -48 + -19x + -1x2

Reorder the terms:
-12 + 12x + 2x2 = 48 + -48 + 19x + -19x + x2 + -1x2

Combine like terms: 48 + -48 = 0
-12 + 12x + 2x2 = 0 + 19x + -19x + x2 + -1x2
-12 + 12x + 2x2 = 19x + -19x + x2 + -1x2

Combine like terms: 19x + -19x = 0
-12 + 12x + 2x2 = 0 + x2 + -1x2
-12 + 12x + 2x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-12 + 12x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-6 + 6x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-6 + 6x + x2)' equal to zero and attempt to solve:

Simplifying
-6 + 6x + x2 = 0

Solving
-6 + 6x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 6x + 6 + x2 = 0 + 6

Reorder the terms:
-6 + 6 + 6x + x2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 6x + x2 = 0 + 6
6x + x2 = 0 + 6

Combine like terms: 0 + 6 = 6
6x + x2 = 6

The x term is 6x.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6x + 9 + x2 = 6 + 9

Reorder the terms:
9 + 6x + x2 = 6 + 9

Combine like terms: 6 + 9 = 15
9 + 6x + x2 = 15

Factor a perfect square on the left side:
(x + 3)(x + 3) = 15

Calculate the square root of the right side: 3.872983346

Break this problem into two subproblems by setting 
(x + 3) equal to 3.872983346 and -3.872983346.

Subproblem 1
x + 3 = 3.872983346

Simplifying
x + 3 = 3.872983346

Reorder the terms:
3 + x = 3.872983346

Solving
3 + x = 3.872983346

Solving for variable 'x'.

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

Add '-3' to each side of the equation.
3 + -3 + x = 3.872983346 + -3

Combine like terms: 3 + -3 = 0
0 + x = 3.872983346 + -3
x = 3.872983346 + -3

Combine like terms: 3.872983346 + -3 = 0.872983346
x = 0.872983346

Simplifying
x = 0.872983346

Subproblem 2
x + 3 = -3.872983346

Simplifying
x + 3 = -3.872983346

Reorder the terms:
3 + x = -3.872983346

Solving
3 + x = -3.872983346

Solving for variable 'x'.

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

Add '-3' to each side of the equation.
3 + -3 + x = -3.872983346 + -3

Combine like terms: 3 + -3 = 0
0 + x = -3.872983346 + -3
x = -3.872983346 + -3

Combine like terms: -3.872983346 + -3 = -6.872983346
x = -6.872983346

Simplifying
x = -6.872983346

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.872983346, -6.872983346}
Solution
x = {0.872983346, -6.872983346}