To solve the given rational equation:
x+4/x+5=6/x^2+10x+25
We will begin by first finding the LCD (Least Common Denominator) which is:
(x+5)(x+5)= (x+5)^2
Multiplying both sides of the equation by the LCD, we get:
(x+4)(x+5)(x+5) = 6(x+5)^2
Expanding both sides of the equation and simplifying further, we get:
x^3 + 14x^2 + 61x + 80 =0
Now, we can use the rational root theorem to identify the possible rational roots of the polynomial equation. The possible rational roots are:
±1, ±2, ±4, ±5, ±8, ±10, ±20, ±40, ±80
Testing these rational roots, we get that x = -4 is a root of the polynomial. Using synthetic division, we can then factor the polynomial as:
(x+4)(x^2 + 10x + 20) = 0
This gives us the solutions:
x = -4, x = -5 + 3sqrt(2)i and x = -5 - 3sqrt(2)i
Therefore, the excluded values are x = -5 and x = -5 ± sqrt(2)i.