Convert two polynomials into a target polynomials through operations You are provided with two polynomials $x^3-3x^2+2$ and $x^2-4x$ on a
piece of paper.
If there are polynomials $f(x)$ and $g(x)$ on the paper, you are allowed
to write $f(x) + g(x)$, $f(x) - g(x)$, $f(x)g(x)$, $f(g(x))$ and
$cf(x)$.
Can you obtain a linear polynomial (exactly one degree) given
two starting polynomials?

I feel like this could be a linear algebra problem, if the operations are just $f(x) + g(x)$, $f(x) - g(x)$ and $cf(x)$ Then we are looking for the linear dependence. But now sure how to handle $f(x)g(x)$, $f(g(x))$
As mentioned, subtraction decreases the degree if the leading coefficients are the same (which we can ensure with $cf(x)$), so all we need is polynomials of the same degree. This is the crux. See if you can work it out before reading on.

We currently do not have 2 polynomials of the same degree, so let's force them out and take their difference. I will work with the specific cases of $a(x) = x^3 - 3x^2 + 2$, $b(x) = x^2 - 4x$, but I suspect this will generalize (with certain conditions).
Step 1: Calculate $a(x)a(x) = x^6 - 6x^5 + 9x^4 + 4x^3 - 12x^2 + 4$.
Step 2: Calculate $b(x)b(x)b(x) = x^6 - 12x^4 + 48x^2 - 64  $. This gives us 2 polynomials of degree 6.
Step 3: Take their difference $a(x)^2 - b(x)^3 = 6x^5 -39 x^4 + 68x^3 - 12x^2 + 4 = c(x) $.
Step 4: Calculate $a(x) b(x) = x^5 - 7x^4 + 12x^3 + 2x^2 - 8x $. This gives us 2 polynomials of degree 5.
Step 5: Take their difference $ c(x) - 6a(x) b(x) = -15x^4 + 86x^3 - 96x^2 + 48x + 4 = d(x)$.
Step 6: Calculate $b(x) b(x)  = x^4 - 8x^3 + 16x^2$. This gives us 2 polynomials of degree 4.
Step 7: Take their difference $d(x) + 15 6 b(x)b(x) = -34x^3 + 144x^2 + 48x + 4 = e(x) $, which is a polynomial of degree 3.
Step 8: Take their difference $ e(x) + 34 a(x) = 42x^2 + 48x + 72 = f(x) $, which is a polynomial of degree (at most, and hopefully) 2.
Step 9: Take their difference $ f(x)  - 42 b(x) = 216x+72 $, which is a polynomial of degree 1.

Notes

As SenZen mentions, if our initial polynomials don't have a linear term, then none of the operations would have led to a linear term. So this isn't always doable.
As implied by SenZen, we might lower the degree by more than one if the ratio of the first few coefficients are equal. This has happened when I was using $ a(b(x)) , b(a(x))$ as my starting polynomials, and I didn't end up with a polynomial of degree 1.
This process is not unique. We could have made different choices for the polynomials of higher degrees.
I might have a calculation error. If so, let me know. This was way to much calculation for a contest-math problem, even after knowing the crux, so I'd like to know of an alternative solution.
No.
The degrees of both polynomials is greater that one and none of those operations decreases the degree.