multiply r^2+7r+10/3 by 3r-30/r^2-5r-50

Whenever you have a fraction, you need to enclose the numerator within parentheses.

The same is true of the denominator.
If you omit the parentheses in an exam, you will be marked wrong.

The first expression will be mathematically interpreted as:
r^2+7r+(10/3) because of the priority of multiplication and division over addition and subtraction.

I assume that the expressions have the parentheses inserted:
(r^2+7r+10)/3 * (3r-30)/(r^2-5r-50)

Factorize each expression:

=(r+5)(r+2)/3 * 3(r-10)/((r-10)(r+5))
Cancel out common factors in italics to leave the answer as:
=(r+2)

To multiply the expression (r^2 + 7r + 10/3) by (3r - 30) / (r^2 - 5r - 50), you can follow these steps:

Step 1: Factor the denominators.
- The denominator of the first fraction, (r^2 - 5r - 50), can be factored as (r - 10)(r + 5).
- The denominator of the second fraction, (r^2 - 5r - 50), can also be factored as (r - 10)(r + 5).

So the expression becomes: (r^2 + 7r + 10/3) * (3r - 30) / ((r - 10)(r + 5)).

Step 2: Simplify the expression.
- Multiply the numerators: (r^2 + 7r + 10/3) * (3r - 30) = (3r^3 - 30r^2 + (10/3)(3r - 30).

Step 3: Expand and combine like terms in the numerator.
- Multiply (10/3) by (3r - 30):
(10/3) * (3r) = 10r;
(10/3) * (-30) = -100.

So, the expanded numerator becomes: 3r^3 - 30r^2 + 10r - 100.

Step 4: Combine the numerator and denominator.
- The expanded numerator is 3r^3 - 30r^2 + 10r - 100.
- The denominator is (r - 10)(r + 5).

Thus, the final expression is: (3r^3 - 30r^2 + 10r - 100) / ((r - 10)(r + 5)).