If you used words in addition to the symbols, your "proof" could become valid. Many students seem to think that words are unnecessary in a proof; in fact, they are essential. (In JeffM's version, logical symbols, which you may not have learned yet, play that role.)
You need to state how one line relates to the lines before it; as written, you have
ASSUMED that the inequality is (always) true, and derived from that a true inequality. That is what you do to
solve an inequality, in this case to discover that it is always true. But that is not the actual proof; what your work shows is that
if the given inequality is true,
then -15 > -35. Here is your work, turned into a proof by explaining the connections:
We want to prove that (a+3)(a-5)›(a-7)(a+5).
This is equivalent to a2-5a+3a-15›a2+5a-7a-35, which is equivalent to a2-2a-15›a2-2a-35.
Subtracting a2-2a from both sides, we find that this is equivalent to -15›-35, which is true independent of the variable a.
Therefore, the original inequality is true for all a.
Reversing the order of statements, as JeffM did, makes a neater proof.
Presumably your teacher is trying to teach you what constitutes a valid proof. Pay attention and learn!
EDIT: Here is another way to turn what you wrote into a valid proof:
We can rewrite (a+3)(a-5)›(a-7)(a+5) as the following equivalent inequality:
a2-5a+3a-15›a2+5a-7a-35.
a2-2a-15›a2-2a-35.
But since
a2=a2;
-2a=-2a;
-15›-35;
we see that the equivalent inequality is always true.
Proven.