  This paper considers various formulations of the sum-product problem. It is
shown that, for a finite set $A\subset{\mathbb{R}}$,
$$|A(A+A)|\gg{|A|^{\frac{3}{2}+\frac{1}{178}}},$$ giving a partial answer to a
conjecture of Balog. In a similar spirit, it is established that
$$|A(A+A+A+A)|\gg{\frac{|A|^2}{\log{|A|}}},$$ a bound which is optimal up to
constant and logarithmic factors. We also prove several new results concerning
sum-product estimates and expanders, for example, showing that
$$|A(A+a)|\gg{|A|^{3/2}}$$ holds for a typical element of $A$.
