Simplify the complex fraction:

[(4/(x-5))+(2/(x+2))]/
[((2x)/(x^2-3x-10))+3]

It was rubbish

To simplify the complex fraction, we need to find a common denominator for both the numerator and denominator.

Let's start by finding the LCD (Least Common Denominator) for the numerator. The denominators in the numerator are (x-5) and (x+2). To find the least common multiple of these denominators, we can multiply them together, but we need to factorize the quadratic expression in the second term of the denominator first.

The quadratic expression is x^2-3x-10. We can factorize it as (x-5)(x+2). So, the factored form of the quadratic expression is (x-5)(x+2).

Now, let's find the LCD by multiplying the two denominators together:
LCD = (x-5)(x+2)

Next, multiply each term in the numerator by the missing factor to get rid of the fractions. Then, combine like terms in the numerator.

Numerator:
[(4/(x-5)) * (x+2) + (2/(x+2)) * (x-5)]
= (4(x+2) + 2(x-5))/(x-5)(x+2)
= (4x + 8 + 2x - 10)/(x-5)(x+2)
= (6x - 2)/(x-5)(x+2)

Now, let's simplify the denominator.
The denominator is [(2x)/(x^2-3x-10)] + 3.

We already factored the quadratic expression as (x-5)(x+2). So we can rewrite the denominator as [(2x)/((x-5)(x+2))] + 3.

Since the first term in the denominator already has a common denominator with the numerator, we can combine the two terms in the denominator into a single fraction by using the common denominator.

[(2x)/((x-5)(x+2))] + 3
= [(2x+3(x-5)(x+2))/((x-5)(x+2))]

Now, we have the simplified complex fraction:
[(6x - 2)/(x-5)(x+2)] / [(2x+3(x-5)(x+2))/((x-5)(x+2))]

To divide by a fraction, we multiply by its reciprocal:
= [(6x - 2)/(x-5)(x+2)] * [((x-5)(x+2))/(2x+3(x-5)(x+2))]

Now, let's cancel out the common factors where possible and simplify the expression further if needed.

putting top and bottom over a common denominator of (x-5)(x+2) and then factoring it out, we have

(6x-2)/(3x^2-7x-30)