I have a math problem and the answer to it. The only problem I have is that I cannot figure out how to get the answer given. I have posted a picture of the math problem below please help me understand this math problem. I would greatly appreciate it.

Note that 5p^2 + 16p - 16 = (p+4)(5p-4), which helps somewhat. Make it a habit to try and factorize quadratics when given this kind of problem to aid in finding easy common denominators.

In this problem, we could make the common denominator of each term (4-5p)(5p-4)(p+4).

For example, this would mean that in the first term the numerator would be 8(5p-4)(p+4).
Do the same for the other two terms.
Combine the terms.
Simplify the numerator.
Divide by monomials ex. (4-5p),(5p-4),(p+4) if possible.

done.

Im still confused could you elaborate a little more?

What would you like elaborated?

here is where im stuck I got to this point I dont know what to do now?

Assuming that your expansions are correct, all that's left to do on the numerator is to collect like terms. In this example, we can reorder the numerator as

40p^2 - 10p^2 -5p^2 + 160p - 32p -8p -40p + 4p + 20p - 128 + 32 - 16 which can be rewritten as
(40 - 10 - 5)p^2 + (160 - 32 - 8 - 40 + 4 + 20)p + (-128 + 32 - 16)

Simplify, and then you're done.

alright so i have combined everything and I am stuck again. 25p^2+104p-112/(4-5p)(5p-4)(p+4) I dont know what to do from here to get the answer am I even close it says the answer is -9p-44/(5p-4)(p+4)

(-9p-44) * (4-5p) = 45p^2 - 36p +140p - 176 = 45p^2 -104p - 176 , so somewhere along the way you did not expand correctly. You are close, however. Try again from scratch and use the same general process I introduced. If you cannot solve it after a meaningful effort, I'll write a full detailed solution.

alright so i have combined everything and I am stuck again. 25p^2+104p-112/(4-5p)(5p-4)(p+4) I dont know what to do from here to get the answer am I even close it says the answer is -9p-44/(5p-4)(p+4)

Idk what to do. Ive been working on this same problem all day for like probably 5 hours now. Could u help me out and give me a detailed solution

Alright, here's a detailed answer. You must have done something wrong in the middle steps, but the concept of common denominator is still the same. Let us consider each term seperately and simplify

$\displaystyle \frac{8}{4-5p} = \frac{8(5p-4)(p+4)}{(4-5p)(5p-4)(p+4)} = \frac{40p^2 -32p + 160p - 128}{(4-5p)(5p-4)(p+4)}$
$\displaystyle \frac{-2}{5p-4} = \frac{-2(4-5p)(p+4)}{(4-5p)(5p-4)(p+4)} = \frac{10p^2 + 32p - 32}{(4-5p)(5p-4)(p+4)}$
$\displaystyle \frac{(p-4)}{(5p-4)(p+4)} = \frac{(p-4)(4-5p)}{(4-5p)(5p-4)(p+4)} = \frac{24p - 5p^2 - 16}{(4-5p)(5p-4)(p+4)}$

Combining the fractions and like terms like we did before we get
$\displaystyle \frac{45p^2 + 184p - 176}{(4-5p)(5p-4)(p+4)}$

While not immediately obvious
$\displaystyle \frac{45p^2 + 184p - 176}{(4-5p)(5p-4)(p+4)} = \frac{-9p-44}{(5p-4)(p+4)}$

since (4-5p) divides (45p^2 + 184p - 176). You can use polynomial long division to verify.

If you have any more questions, let me know. I still want you to practice your skills and challenge your knowledge base so you are ready for exams EDIT:// computation error.

Code:
     p - 4            8          2
=============== - ======== - ========
(p + 4)(5p - 4)    5p - 4     5p - 4

p - 4      8     2
======== - === - ===
k(p + 4)    k     k
Since 8/(4 - 5p) = -8/(5p - 4), rearrange terms as I show above;
then, to keep "as simple as possible", let k = 5p - 4 (as above).

Now do the "work", then replace k by 5p - 4 : get my drift?