algebra help finding all solutions?

(x)/(2x+7) - (x+1)/(x+3) = 1

x / (2x + 7) = 1 + [(x + 1) / (x + 3)

x / (2x + 7) = [(x + 3) + (x + 1)] / (x + 3)

x / (2x + 7) = (2x + 4) / (x + 3)

x (x + 3) = (2x + 4) (2x + 7)

x² + 3x = 4x² + 14x + 8x + 28

x² - 4x² + 3x - 22x - 28 = 0

-3x² - 19x - 28 = 0

3x² + 19x + 28 = 0

3x² + 12x + 7x + 28 = 0

3x (x + 4) + 7 (x + 4) = 0

(3x + 7) (x + 4) = 0

x = -7/3 OR x = -4 ANSWER.

Multiply all terms by (x+3)(2x+7) which will leave you with...
(x+3)(x) - (x+1)(2x+7) = (x+3)(2x+7)
x^2 + 3x - (2x^2+9x+7) = 2x^2 + 13x + 21
x^2 + 3x - 2x^2 - 9x - 7 = 2x^2 + 13x + 21
-x^2 - 6x - 7 = 2x^2 + 13x + 21
0 = 3x^2 + 19x + 28
0 = (3x+7)(x+4)

x = -7/3 or -4

Multiply eqution by (x+3)(2x+7)
(x+3)(x) - (x+1)(2x+7) = (x+3)(2x+7)
x^2 + 3x - (2x^2+9x+7) = 2x^2 + 13x + 21
x^2 + 3x - 2x^2 - 9x - 7 = 2x^2 + 13x + 21
-x^2 - 6x - 7 = 2x^2 + 13x + 21
0 = 3x^2 + 19x + 28
0 = (3x+7)(x+4)
0 = (3x+7) or 0 = (x+4)
3x= -7
x=-7/3 or x= -4
solution is x={-7/3,-4}

(x/[2x + 7]) - ([x + 1]/[x + 3]) = 1
([x² + 3x] - [2x² + 2x + 7x + 7])/(2x² + 6x + 7x + 21) = 1
(x² + 3x - [2x² + 9x + 7])/(2x² + 13x + 21) = 1
x² + 3x - 2x² - 9x - 7 = 2x² + 13x + 21
- x² - 6x - 7 = 2x² + 13x + 21
3x² + 19x = - 28
x² + 19/3x = - 28/3
x² + 19/6x = - 28/3 + (19/6)²
x² + 19/6x = - 336/36 + 361/36
(x + 19/6)² = 25/36
x + 19/6 = 5/6

x = 5/6 - 19/6, x = - 14/6, x = - 7/3
x = - 5/6 - 19/6, x = - 24/6, x = - 4

Answer: x = - 7/3, - 4

Proof (x = - 7/3):
([- 7/3]/[2{- 7/3} + 7]) - ([- 7/3 + 1]/[- 7/3 + 3]) = 1
([- 7/3]/[- 14/3 + 21/3]) - ([- 7/3 + 3/3]/[- 7/3 + 9/3]) = 1
([- 7/3]/[7/3]) - ([- 4/3]/[2/3]) = 1
- 1 - (- 2) = 1
- 1 + 2 = 1
1 = 1

Proof (x = - 4):
(- 4/[2{- 4} + 7) - ([- 4 + 1]/[- 4 + 3]) = 1
(- 4/[- 8 + 7) - (- 3/- 1) = 1
(- 4/- 1) - (3) = 1
4 - 3 = 1
1 = 1

First of all we know the values that x is not equal to by virtue of the denominators. We cannot divide by zero:
x≠-3, x≠-7/2
(x)/(2x+7) - (x+1)/(x+3) = 1
(x)*(x+3) - (x+1)*(2x+7) = 1*(x+3)*(2x+7)
x^2 +3x - 2x^2 -9x -7 = 2x^2 +13x + 21
Combine like terms and solve for x. Whatever solutions you get except those excluded above will work. I leave it to you to finish the problem.

x/(2x + 7) - (x + 1)/(x + 3) = 1
[2x + 7][x + 3][x/(2x + 7) - (x + 1)/(x + 3)] = [2x + 7][x + 3][1]
x(x + 3) - (x + 1)(2x + 7) = (2x + 7)(x + 3)
x^2 + 3x - (2x^2 + 2x + 7x + 7) = 2x^2 + 7x + 6x + 21
x^2 + 3x - 2x^2 - 2x - 7x - 7 = 2x^2 + 13x + 21
x^2 - 2x^2 + 3x - 9x - 7 = 2x^2 + 13x + 21
-x^2 - 6x - 7 = 2x^2 + 13x + 21
-x^2 - 2x^2 - 6x - 13x - 7 - 21 = 0
-3x^2 - 19x - 28 = 0
3x^2 + 19x + 28 = 0
3x^2 + 12x + 7x + 28 = 0
(3x^2 + 12x) + (7x + 28) = 0
3x(x + 4) + 7(x + 4) = 0
(x + 4)(3x + 7) = 0

x + 4 = 0
x = -4

3x + 7 = 0
3x = -7
x = -7/3

∴ x = -4 , -7/3

x/(2x+7)=x+1/x+3 +x+3/x+3=(2x+4)/(x+3)=========>
x^2+3x=4x^2+8x+14x+28,
3x^2+19x+28=0,
x=-4 & x=-7/3