Can math be subjective? Often times in math, ever since Kindergarten and before, math has been defined by the fact that there are only one answer for problems. For example: $1+1=2$ and $\frac{d}{dx}x^2=2x$. What I am showing by these two examples are two questions that are from completely different areas of math. However, they both have have only one solution. Problems with multiple answers doesn't necessarily mean they are subjective though, such as $|x|=2,$ which has two solutions. My question is that are any such problems that depend entirely on perspective? If all subjective math problems follow a certain pattern, please tell me what that pattern is. I really have no idea of any examples of this and I would really be interested to see one. Thank you very much.
There's plenty of room for subjective opinion in mathematics. It usually doesn't concern questions of the form Is this true? since we have a good consensus how to recognize an acceptable proof and which assumptions for such a proof you need to state explicitly.

As soon as we move onwards to Is this useful? and Is this interesting?, or even Is this likely to work?, subjectivity hits us in full force. Even in pure mathematics, it's easy to choose a set of axioms and derive consequences from them, but if you want anyone to spend time reading your work, you need to tackle the subjective questions and have an explanation why what you're doing is either useful or interesting, or preferably both.

In applied math, these questions are accompanied by Is this the best way to model such-and-such real-world problem? -- where "best" again comes down to usefulness (does the model answer questions we need to have answered?) and interest (does the model give us any insight about the situation we wouldn't have without it?).

The subjective questions are important in research, but can also arise at more elementary level. The high-school teacher who chooses to devote several lessons to presenting Cardano's method for solving the generic third-degree equation will certainly have to answer his students' questions why this is useful or interesting. Perhaps he has an answer. Perhaps he has an answer that the students don't agree with. In that case, he cannot look for a deductive argument concluding that Cardano's formula is interesting -- he'll have to appeal to emotions, curiosity, all of those fluffy touchy-feely considerations that we need to use to tackle subjective questions.
Yeah, it can:


Consider that polynomials of degree 4 or lower are said to be solvable (or solvable "in radicals" or whatever) whereas the higher-order ones are said to be unsolvable. But this is quite subjective and arbitrary. The definition of $\sqrt{2}$ is literally "the positive real number $x$ that satisfies $x^2 = 2$". What makes solving $x^2 = 2$ any different from solving $x^5 + x + 10 = 0$? I really don't see a meaningful distinction -- both are real numbers and both are irrational. To write either of them out as decimals, you have to use the same kinds of root-finding algorithms for both. To me, the distinction that one of them can be written in terms of radicals and the other one can't seems pretty arbitrary and subjective.
Just look at the definition of e.g. "closed-form solution". It's completely subjective.


etc.