Question 697214
{{{ f(x)=x^3 +5x^2 +17x +13 }}}


If x=-1 is a root then {{{ (x+1) }}} is a factor so
{{{ f(x)=(x+1)(ax^2 + bx +c) }}}

By observation we can see the coefficient of {{{ x^3 }}} is 1 so {{{ a = 1 }}} so:
{{{ f(x)=(x+1)(x^2 + bx +c) }}}
{{{ f(x)=(x^3 + bx^2 +cx) + (x^2 + bx +c) }}}
{{{ f(x)=x^3 + (b+1)x^2 + (c+b)x + c }}}


Comparing coefficients of {{{ x^0 }}} : {{{ c=13 }}}
Comparing coefficients of {{{ x^2 }}} : {{{ b+1=5 }}} so {{{ b=4 }}}
Comparing coefficients of {{{ x^1 }}} : {{{ c+b=17 }}} and is consistent with above



Hence, 
{{{ f(x)=(x+1)(x^2 + 4x + 13) }}}


This does not factorize further