Question

x^4+10x^3+35x^2+50x+24

Answer 1

Put x = ( -1 ) [ Hit and Trial ]

We get : p( -1 ) = 0

By factor theorem, use ( x - (-1) ) as a factor and rewrite :
$p_{1}(x) = {x}^{3} + 9 {x}^{2} + 26x + 24$

Repeat the same step for : x = -2 and rewrite
$p_{2}(x) = {x}^{2} + 7x + 12$
Solve the quadratic and get the factors :
$(x + 3)(x + 4)$
Collect all the factors and rewrite :
$p(x) = (x + 1)(x + 2)(x + 3)(x + 4)$
This is the required factorization

However =_= it does seem like you need the roots of your equation...

Equate everything with zero and you get... either of the factors equals to zero and hence :

The roots of the equation are : x = { -1 , -2 , -3 , -4 }