To simplify the given fraction [tex]\(\frac{2u^2 - 26u + 80}{u^2 - 4u - 5}\)[/tex], we will follow these steps:
1. Factorize the numerator and the denominator (if possible).
Let's start with the numerator: [tex]\(2u^2 - 26u + 80\)[/tex].
We can look for two numbers that multiply to [tex]\(2 \times 80 = 160\)[/tex] and add to [tex]\(-26\)[/tex]. These two numbers are [tex]\(-16\)[/tex] and [tex]\(-10\)[/tex].
So, we can split the middle term and rewrite the numerator:
[tex]\[2u^2 - 26u + 80 = 2u^2 - 16u - 10u + 80\][/tex]
Next, we factor by grouping:
[tex]\[= 2u(u - 8) - 10(u - 8)\][/tex]
[tex]\[= (2u - 10)(u - 8)\][/tex]
Remember to factor out [tex]\(2\)[/tex] from [tex]\(2u - 10\)[/tex] to simplify it further:
[tex]\[= 2(u - 5)(u - 8)\][/tex]
Now, let's factor the denominator: [tex]\(u^2 - 4u - 5\)[/tex].
Similarly, we look for two numbers that multiply to [tex]\(-5\)[/tex] and add to [tex]\(-4\)[/tex]. These two numbers are [tex]\(-5\)[/tex] and [tex]\(1\)[/tex].
So, we can factor the quadratic directly:
[tex]\[u^2 - 4u - 5 = (u - 5)(u + 1)\][/tex]
2. Substitute the factored forms into the original fraction.
Now our fraction looks like:
[tex]\[
\frac{2(u - 5)(u - 8)}{(u - 5)(u + 1)}
\][/tex]
3. Cancel out common factors from the numerator and the denominator.
We see that [tex]\((u - 5)\)[/tex] is a common factor in both the numerator and the denominator. Therefore, we can cancel [tex]\((u - 5)\)[/tex]:
[tex]\[
\frac{2(u - 8)}{u + 1}
\][/tex]
4. Write the final simplified fraction.
After canceling the common factors, we are left with:
[tex]\[
\frac{2(u - 8)}{u + 1}
\][/tex]
So, the simplified form of the given fraction is:
[tex]\[
\boxed{\frac{2(u - 8)}{u + 1}}
\][/tex]
