The study of the maximum number of limit cycles for the Liénard equation \(x'=y,\quad y'=-g(x)-f(x)y\), when \(f\) and \(g\) are polynomials of low degree is an interesting and yet open problem. Here, the authors study the number and configuration of limit cycles of the above equation when \(f(x)= (x-c)(x-d),\) \(g(x)=x(x-a)(x+b)\) and \(a,b,c,d\) are real parameters satisfying \(a>0, b>0\) and \(d>c\). Although they are not able to solve completely the problem their study covers a lot of cases. Observe that this Liénard system has exactly three critical points, one saddle and two antisaddles. Hence, the system can have two types of limit cycles, the ones surrounding just one antisaddle (called in the paper small limit cycles) and the ones surrounding the three singularities (called large limit cycles). The methodology goes as follows: they consider each one of the possible ordering of the numbers \(\{a,-b,c,d\},\) assuming that all them are different (these cases are called generic in the paper) and afterwards each one of the cases in which some zero of \(g\) coincides with some zero of \(f\) (nongeneric cases). Then, for each one of these cases, they try to apply either known results on Liénard systems (for instance results of Rychkov or Zhang) or to create new criteria. They get their results doing this case by case study firstly for small limit cycles and afterwards for large limit cycles. For instance, one of the remaining open problems is the following: Prove that there are at most 2 limit cycles surrounding the point \((-b,0)\) when \(-b<c<d<0<a.\)