We need to use rational root theorem to find out roots here.
The rational root theorem states that if p(x) is a polynomial with integer coefficients and if  is a zero of p(x) then p is a factor of constant term and q is a factor of leasing term coefficient.
 is a zero of p(x) then p is a factor of constant term and q is a factor of leasing term coefficient.
Here factors of constant term are 1,2,3,4,6,8,12,24,-1,-2,-3,-4,-6,-8,-12, and -24.
And factors of leading coefficient is -1,1.
Hence possible roots may be -1,1,-2,2,-3,3,-4,4,-6,6,-8,8,-12,12,-24 and 24.
Let us plugin these in f(x) to find zeroes.

Hence x=-1 is a zero which means x-(-1)=x+1 is a factor.
Let us use synthetic division to find quotient.
-1 | 1  10  35  50  24
   <u>| 0  -1  -9  -26   -24</u>
    <u> 1    9  26  24    0</u>
Hence quotient is 
Since all coefficients are positive, root must be negative. Let's plugin all remaining negative numbers in the quotient.

Hence x+2 is another factor.
Let us find quotient again using synthetic division.
-2 | 1   9  26   24
    <u>| 0  -2  -14   -24</u>
    <u>  1    7    12     0</u>
Hence quotient is 
Again we got quotient with all positive coefficients, let us plugin remaining negative numbers from rational root theorem.

Hence x+3 is also a factor.
Let us find quotient using synthetic division.
-3 | 1  7  12
    <u>| 0 -3  -12</u>
     <u> 1    4    0</u>
Hence quotient is x+4.
So, 
Please have a look at the graph attached.