Given the function f(x)=x^4+10x^3+35x^2+50x+24 , factor completely. Show all work and steps. Then sketch the graph.

According to the rational root theorem, the possible rational factors, if any, of the given polynomial are the factors of 24/1=24.
Thus possible factors are
(x +/- k) where k=1,2,3,4,6,8,12,24.
Since all terms are positive, (x-k) do not exist as factors (see Descartes rule of signs) for all k.
Of the remainder, we note that the
sum of coefficients of even degree terms=1+35+24=60
sum of coefficients of odd degree terms=10+50=60
This means that (x+1) is a factor.

Divide f(x) by (x+1) =>
g(x)=f(x)/(x+1)=x^3+9x^2+26x+24 (no remainder)

We continue with x+2=0, or put x=-2 into g(x), and get
g(-2)=-8+36-52+24=0  => (x+2) is another rational factor.

Again, divide g(x) by (x+2) to get
h(x)=x^2+7x+12 (remainder=0)

We can readily factor h(x) by inspection as h(x)=(x+3)(x+4)
[from 3+4=7,3*4=12]

Therefore

f(x)=x^4+10*x^3+35*x^2+50*x+24=(x+1)(x+2)(x+3)(x+4)