Write $f(x) =x^3+x^2-3x-3$ as a product of a linear factor and a quadratic factor I know that the linear factor is $(x+1) (x^2-3),$ but how would I find the quadratic factor?

Simply calculate the roots of your polynomial (if possible).
You have a polynomial of degree 3. 
Normally you first try to find a root, by "guessing". We have to check the divesors of the constant -3. Which are $\pm 3, \pm 1$.
Easily we see, that x=-1 is a root and then make a long division
$(x^3+x^2-3x-3)\div (x+1)$ giving us the quadratic factor.
On other way is to compare coefficients by:
$(x^3+x^2-3x-3)=(ax+b)(cx^2+dx+e)$
Which should involve more calculation.
