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Usually simple substitutions enable such expressions to be seen as quadratic expressions.
The substitutions x = a2 and y = b2 give
a4 + b4 - 7a2b2 = x2 - 7xy + y2 which does not have any rational factors.
Consequently, the quartic in a and b does not have rational factors.
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Unfortunately, limitations of the browser used by WA means that we cannot see most symbols. It is therefore impossible to give a proper answer to your question. Please resubmit your question spelling out the symbols as "plus", "minus", "equals" etc.
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