Remark
This one isn't easy. There are a lot of steps.
Step One
Multiply the entire equation through by 4*(x - 1)(x + 2)
4*(x - 1)(x + 2)[1/(x - 1) + 1/(x + 2) = 5/4]
4(x +2) + 4(x - 1) = 5*(x + 2)(x - 1) If you don't see how I got this step, leave a note. Remove the brackets on the left
4x + 8 + 4x - 4 = 5(x +2)(x - 1)
8x + 4 = 5(x + 2)(x - 1) Expand the right side. Use FOIL
8x + 32 = 5 (x^2 + 2x - x - 2)
8x + 32 = 5( x^2 +x - 2) Remove the brackets on the right.
8x + 32 = 5x^2 + 5x - 10 Subtact 8x + 32 from both sides.
0 = 5x^2 + 5x - 8x - 32 - 10
0 = 5x^2 - 3x - 42
Now you have to factor this mess. You need two numbers that multiply to 5 which can only be 5 and 1. You also need 2 numbers which multiply to 42. That's not so easy. 6 and 7 are your best bet because you need something close to get 3.
It turns out that none of these numbers work. So you have to use the quadratic formula
a = 5
b = - 3
c = - 42
[tex] \text{x = }\dfrac{ -b \pm \sqrt{b^{2} - 4ac } }{2a}[/tex]
[tex] \text{x = }\dfrac{ -3 \pm \sqrt{(-3)^{2} - 4(1)(-42)} }{10} [/tex]
[tex] \text{x = }\dfrac{ -3 \pm \sqrt{9 + 840} }{10} [/tex]
x = 3.12
x = -2.61
