Question

first up, lets review the assignments learning objectives
get familiar with this topic by reviewing instruction and answering a couple of questions.

question
rewrite as equivalent rational expressions with denominator \\((5u + 5)(u - 9)(u + 5)\\):
\\\frac{9}{5u^2 - 40u - 45}, \frac{2u}{5u^2 + 30u + 25}\\

provide your answer below:

Explanation:

Factor the first denominator

To find equivalent expressions, we first factor the denominator of the first rational expression, \(5u^2 - 40u - 45\).
Using the Greatest Common Factor Factoring and Trinomial Factorization knowledge points:

$$ 5u^2 - 40u - 45 = 5(u^2 - 8u - 9) $$

Factoring the quadratic trinomial inside the parentheses:

$$ 5(u^2 - 8u - 9) = 5(u - 9)(u + 1) = (5u + 5)(u - 9) $$

Factor the second denominator

Next, we factor the denominator of the second rational expression, \(5u^2 + 30u + 25\).
Using the Greatest Common Factor Factoring and Trinomial Factorization knowledge points:

$$ 5u^2 + 30u + 25 = 5(u^2 + 6u + 5) $$

Factoring the quadratic trinomial inside the parentheses:

$$ 5(u^2 + 6u + 5) = 5(u + 5)(u + 1) = (5u + 5)(u + 5) $$

Rewrite the first expression

The target denominator is \((5u + 5)(u - 9)(u + 5)\).
The factored form of the first denominator is \((5u + 5)(u - 9)\).
To obtain the target denominator, we multiply both the numerator and the denominator by the missing factor, \((u + 5)\):

$$ \frac{9}{(5u + 5)(u - 9)} \cdot \frac{u + 5}{u + 5} = \frac{9(u + 5)}{(5u + 5)(u - 9)(u + 5)} $$

Rewrite the second expression

The factored form of the second denominator is \((5u + 5)(u + 5)\).
To obtain the target denominator, we multiply both the numerator and the denominator by the missing factor, \((u - 9)\):

$$ \frac{2u}{(5u + 5)(u + 5)} \cdot \frac{u - 9}{u - 9} = \frac{2u(u - 9)}{(5u + 5)(u - 9)(u + 5)} $$

Answer:

$$\frac{9(u + 5)}{(5u + 5)(u - 9)(u + 5)}, \frac{2u(u - 9)}{(5u + 5)(u - 9)(u + 5)}$$