Transform this formula so that all the quantifiers are located at the beginning of the formule:
$\big((\exists{x})(a(x) \Rightarrow b(x)\big) \Rightarrow \big((\exists{x})a(x) \Rightarrow (\exists{x})b(x)\big) $
Transform this formula so that all the quantifiers are located at the beginning of the formule:
$\big((\exists{x})(a(x) \Rightarrow b(x)\big) \Rightarrow \big((\exists{x})a(x) \Rightarrow (\exists{x})b(x)\big) $
Note: Each existential quantifier in your statement is limited in its scope.
$((\exists{x})(a(x) \Rightarrow b(x)) \Rightarrow ((\exists{x})a(x) \Rightarrow (\exists{x})b(x)) \tag{1}$
The following statement is equivalent to $(1)$:
$((\exists{x})(a(x) \Rightarrow b(x)) \Rightarrow ((\exists{y})a(y) \Rightarrow (\exists{z})b(z)).\tag{2}$
The quantified variables first need to be disambiguated in $(1)$, as shown in $(2)$ - to transform $(1)$ into a statement with all quantifiers at the start of the statement.
I'll let you take $(2)$ from here. Most of the work is simply keeping straight what's being quantified, and where.
Care must be taken with respect to quantifiers that appear in the antecedent of an implication, when transforming such statements so all quantifiers appear at the start of the entire statement. In this case, it turns out, that the final representation leaves the quantifiers unchanged.
But as you'll see with your more complicated case (the one you posted in your final comment below, that's not always the case.
EDIT: In response to the work shown in the comments below (user please do not delete comments below):
Slow down a bit: From $(2)$, and given the question you posted above, we can, if need be, bring all the quantifiers to the start, as there three existentially quantified variables. $(\exists x)(\exists y)(\exists z)\left((a(x) \Rightarrow b(x)) \Rightarrow (a(y) \Rightarrow b(z))\right).\tag{3}$
Here's something more than a mere hint, but less than a full answer doing your homework for you!