jmc

By Vieta's formulas, $$t=-(a+b)(b+c)(c+a).$$From the first cubic polynomial, we have $a+b+c=-3$. Using this equation, we can rewrite the expression for $t$ as $$t=-(-3-c)(-3-a)(-3-b).$$To compute this expression quickly, notice that, for any $x$, $$x^3+3x^2+4x-11=(x-a)(x-b)(x-c)$$by the factor theorem. Setting $x=-3$, we get $$(-3{)}^3+3(-3{)}^2+4(-3)-11=-23=(-3-a)(-3-b)(-3-c).$$Thus, $t=-(-23)=\boxed{23}$.