could someone help me understand this math problem?

Question

Reduce the rational expression to lowest terms. If it is already in lowest terms, enter the expression in the answer box.

 

(2r²+7r+3)/(r⁴−81)

Answer 1

Since you are not sure whether it's in lowest term or not, you have no choice but to expand.

 

First in the numerator, either by factoring or using the quadratic formula, you will get:

(2r+1)(r+3)

 

In the denominator, you can apply the expansion of difference of two squares, a² - b² = (a+b)(a-b).

 

First, the r⁴ - 81 = (r² + 9)(r² - 9). Then we're gonna expand the (r² - 9) further = (r+3)(r-3);

 

Making r⁴ - 81 = (r² + 9)(r+3)(r-3)

 

Looking at the new fraction: 

 

(2r+1)(r+3)/(r²+9)(r+3)(r-3), you can cancel out (r+3) and it will leave you:

 

(2r+1)/(r²+9)(r-3) 

 

From here, I don't know what your teacher want, whether leave it like that or you should multiply the binomials. However, I think you can take it from here and I hope this helped. :)