Q:

Was the word 'racist' coined by Trotsky in 1927?

According to the image below, "racist is a made up word by Leon Trotsky in 1927."
I searched in the Online Etymology Dictionary and found that

racist (n.)
1932 [as a noun], 1938 as an adjective, from race (n.2); racism is first attested 1936 (from French racisme, 1935), originally in the context of Nazi theories. But they replaced earlier words, racialism (1871) and racialist (1917), both often used early 20c. in a British or South African context. In the U.S., race hatred, race prejudice had been used, and, especially in 19c. political contexts, negrophobia.

So the Online Etymology Dictionary doesn't mention that Trotsky coined the word.
Is the Trotsky—racist theory is true?

A:

According to NPR's Codeswitch, the term racism meaning "discrimination or prejudice based on race", was used before 1927. 
The  term as used in the picture you show seems to have a different connotation  from the more common one: 

The Oxford English Dictionary's first recorded utterance of the word racism was by a man named Richard Henry Pratt in 1902. 

Pratt was railing against the evils of racial segregation.

Segregating any class or race of people apart from the rest of the people kills the progress of the segregated people or makes their growth very slow. Association of races and classes is necessary to destroy racism and classism.

Although Pratt might have been the first person to inveigh against racism and its deleterious effects by name, he is much better-remembered for a very different coinage: Kill the Indian...save the man.
"A great general has said that the only good Indian is a dead one," Pratt said. "In a sense, I agree with the sentiment, but only in this: that all the Indian there is in the race should be dead. Kill the Indian in him, and save the man."
We're still living with the after-effects of what Pratt thought and did. His story serves as a useful parable for why discussions of racism remain so deeply contentious even now.

A:

Although he did not coin it, he may have popularized it afterwards. Trotksy published What is National Socialism in June 1933. I have made racism bold

The theory of race, specially created, it seems, for some pretentious self-educated individual seeking a universal key to all the secrets of life, appears particularly melancholy in the light of the history of ideas. In order to create the religion of pure German blood, Hitler was obliged to borrow at second hand the ideas of racism from a Frenchman, Count Gobineau [4], a diplomat and a literary dilettante. Hitler found the political methodology ready-made in Italy, where Mussolini had borrowed largely from the Marxist theory of the class struggle. Marxism itself is the fruit of union among German philosophy, French history, and British economics. To investigate retrospectively the genealogy of ideas, even those most reactionary and muddleheaded, is to leave not a trace of racism standing.
On the plane of politics, racism is a vapid and bombastic variety of chauvinism in alliance with phrenology. As the ruined nobility sought solace in the gentility of its blood, so the pauperized petty bourgeoisie befuddles itself with fairy tales concerning the special superiorities of its race….
Through the back door, racism returns to economic liberalism, freed from political liberties.

Source: https://www.marxists.org/archive/trotsky/germany/1933/330610.htm
Finally, Trotsky, in Chapter 1 The Peculiarities of Russia's Development of The History of the Russian Revolution (1930), also uses racist with a capital and denigrates the homogeneity of a culture/race:

Slavophilism, the messianism of backwardness, has based its philosophy upon the assumption that the Russian people and their church are democratic through and through, whereas official Russia is a German bureaucracy imposed upon them by Peter the Great. Mark remarked upon this theme: “In the same way the Teutonic jackasses blamed the despotism of Frederick the Second upon the French, as though backward slaves were not always in need of civilised slaves to train them.” This brief comment completely finishes off not only the old philosophy of the Slavophiles, but also the latest revelations of the “Racists.”

Source:https://www.marxists.org/archive/trotsky/1930/hrr/ch01.htm
Furthermore, around the late 1930s (specifically 1937), a rise begins for the terms racist and racism but did not take off until the 1960s.
Source: https://books.google.com/ngrams/graph?content=racism%2C+racist&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cracism%3B%2Cc0%3B.t1%3B%2Cracist%3B%2Cc0
Can it be linked directly to Trotksy? Maybe, considering Soviet Russia had given a lot of resources to subversion, infiltration, and propaganda and had operatives white-washing Russia in American universities and spreading communism to student and some professors, both of which probably sought out reading material of communism, which would lead to Marx, then Lenin, then Trotsky, and so on and so forth.
As for the translation issue. Here is the link to Russian: http://www.magister.msk.ru/library/trotsky/trotl007.htm
Russian:

Славянофильство, мессианизм отсталости, строило свою философию на том, что русский народ и его церковь насквозь демократичны, а официальная Россия -- это немецкая бюрократия, насажденная Петром. Маркс заметил по этому поводу: "Ведь точно так же и тевтонские ослы сваливают деспотизм Фридриха II и т. д. на французов, как будто отсталые рабы не нуждаются всегда в цивилизованных рабах, чтобы пройти нужную выучку". Это краткое замечание исчерпывает до дна не только старую философию славянофилов, но и новейшие откровения "расистов".

According to wiktionary, the word расистов is a translation of racist:
http://en.wiktionary.org/wiki/racist#Translations
http://en.wiktionary.org/wiki/расист#Russian
The capitalization of the word was probably the result of the translator.

As further indication that Trotsky may have helped popularize the term greatly in 1933, a clear correlation can be observed using Google nGram (note that the stated invention of the word in another answer in 1902 was NOT in any way impactful, all the way till late 1920s and into 1930s):

A:

We find the adjective racist penned by Gaston Mery in the November 18, 1897 issue of La Libre Parole "It is time, in popular meetings, that truly French - truly racist- voices oppose their eloquence to the rhetoric of internationalist boastings"

The above passage is from the chapter Birth, Functions and Avatars of the word Racism in the book The Force of Prejudice: On Racism and Its Doubles
The book also quotes Charles Maurras as saying 25 March 1895:

I am myself a 'racist'! Elsewhere I had occasion to say to my distinguished colleague M. Gaston Mery, who at a point became the Knight of the Race and coined the epithet 'racist'. Like him I believe that there is a French race.      

See this book for a more complete history of the word "racist".  
No, it was not coined by Trotsky 1927, it was coined by M. Gaston Mery by 1895.  
See also the 1923 use:

A campaign was even planned to expel from the Nationalist party the agitators of the extreme Right known as “Germanists" or “racists"

Quoting from a publication of the Carnegie Endowment for International Peace at page 53

