Other than using られる with 一段 verbs and changing 五段 into special potential verbs, you can also use できる in the pattern ～ことができる. できる more or less correlates to the potential form of する. As you can see, though, they do not come from the same word.
する’s potential form is できる・出来る. You don't have to add or do anything else. In Japanese, you can use できる with things regarding ability. So, the equivalent of "can English" would be a valid expression.
Together with effort and luck, I was able to be victorious.
Mr. Kato can play go.
I can speak Japanese.
Literally: I can do Japanese.
Grammar Note: You don't make a verb a noun and use "Noun + ができる (see ことができる). This example shows how できる can show ability/talent. So, ゴルフができない would also be correct.
They can't even agree or disagree.
Can I stay for one more day?
Meanings Other Than Potential
The 可能形 is unable to show appearance or what something is made of, but できる sure can. Those are the "other" usages of it mentioned at the beginning of this lesson.
Water is made up of oxygen and hydrogen.
Dinner is made!
8. 人生って、悩みと苦しみからできてるんだなあ。 （男性語）
Life really is made from worries and suffering.
Although humans are made up of the body and the heart, everyone focuses solely on their bodies and forget about providing nurture to their hearts.
Instead of using ～（ら）れる, you may use ことができる. This pattern is arguably more common despite being longer. Without qualification, it will sound as if you can't do the phrase at all when used in the negative. This does not have the requirement of being only used with verbs of volition. This can be used to show the potential of something to happen.
I can speak Japanese.
11. コンピューターを使うことができるか。 （Rough)
Can you use a computer?
I was able to see the American football game.
You won't be able to collect over a million signatures.
I could not read that Kanji.
Particle Note: が in this pattern, due to its specific nature, would otherwise make it sound like you in particular are the one that "can"; however, this is not necessarily the case. Nowadays, ～ことができる has more or less become an affirmative, refined pattern.
～することができる shouldn't be used excessively, and it should be avoided in promotion-oriented settings. However, it shows up a lot in magazines and such because it is felt by many to be more polite. Yet, when there is a word limit, できる should do. For a minority of speakers, this phrase is ungrammatical. However, this is an extreme position based mainly on a misconception of case marking in Japanese. In the potential, が does not mark a subject. It marks an object. This may be confusing, but other languages have similar alignment rules. This is simply a quirk in Japanese, and it's not like natives were confused on how to deal with it until being introduced with half-baked knowledge about Western grammar terms.
You can download it!
Other Ways to Make する Potential
There are four potential routes that you can take to make the potential with する-verbs.
Verb Note: Verbs like 愛す are treated as 五段 verbs. Also note that something like 理解することができる is typically deemed to be unnecessarily wordy.
Curriculum Note: See Potential III for ～得る.
He's a guy that can't sense danger, isn't he?
I can't love you.
To be able to experience.
He's a person that can love anybody, isn't he?
Believe it or not, I couldn't understand Japanese in junior high.