The first step is to factorise the quadratic expression on the right side of the equation. The expression is
x^2 + 9x + 20
We would find two terms such that their sum or difference is 9x and their product is 20x^2. The terms are 5x and 4x. Replacing 9x with 5x and 4x, it becomes
x^2 + 5x + 4x + 20
By factorising, it becomes
x(x + 5) + 4(x + 5)
Since x + 5 is common, it becomes
(x + 4)(x + 5)
Thus, the original expression becomes
x/(x + 4) + 3/(x + 5) = (x + 2)/(x + 4)(x + 5)
The lowest common multiple of the denominators on both sides of the equations is (x + 4)(x + 5). We would multiply each term in the equation by
(x + 4)(x + 5). It becomes
(x + 4)(x + 5)x/(x + 4) + 3(x + 4)(x + 5)/(x + 5) = (x + 2)(x + 4)(x + 5)/(x + 4)(x + 5)
By cancelling out common terms in the numerator and denominator, we have
x(x + 5) + 3(x + 4) = x + 2
We would expand the parentheses on both sides by multiplying the terms inside with the term outside. It becomes
x^2 + 5x + 3x + 12 = x + 2
By collecting like terms, we have
x^2 + 5x + 3x - x + 12 - 2 = 0
x^2 + 7x + 12 = 0
Again, We would find two terms such that their sum or difference is 7x and their product is 12x^2. The terms are 4x and 3x. Replacing 7x with 4x and 3x, it becomes
x^2 + 4x + 3x + 12 = 0
By factorising, it becomes
x(x + 4) + 3(x + 4) = 0
Since x + 4 is common, it becomes
(x + 3)(x + 4) = 0
x + 3 = 0 or x + 4 = 0
x = - 3 or x = - 4
The solutions are x = - 3 or x = - 4
