Simplify the rational expression.

numerator 3 x to the 2 power plus x negative 10 divided by denominator 3 x to the 2 power negative 2x negative 5

Please type these and the others in this fashion:

(3x^2 + x - 10)/(3x^2 - 2x - 5)

= (3x-5)(x+2)/[(3x-5)(x+1)]
= (x+2)/(x+1) , x not equal to 5/3

To simplify the rational expression, we need to simplify the numerator and denominator separately and then simplify the whole expression by canceling out common factors if possible.

Let's start with simplifying the numerator:
Numerator: 3x^2 + x - 10

Since there are no like terms to combine, the numerator is already simplified.

Now let's simplify the denominator:
Denominator: 3x^2 - 2x - 5

To factor this quadratic expression, we need to find two numbers whose product is -5 and whose sum is -2. The numbers that satisfy this condition are -5 and +1, so we can factor the denominator as:

Denominator: (3x - 5)(x + 1)

Now that we have simplified both the numerator and the denominator, we can rewrite the rational expression as:

(3x^2 + x - 10) / (3x - 5)(x + 1)

Therefore, the simplified rational expression is (3x^2 + x - 10) / (3x - 5)(x + 1).