The expression you provided is:

$x^4 + 3x - 4$

This is a polynomial of degree 4, which means it has four roots or solutions in terms of $x$. Finding the roots of a fourth-degree polynomial without specific tools can be challenging, as there is no simple factorization due to the nature of the expression.

Steps to approach this polynomial:

1. Check for Rational Roots: We can start by using the Rational Root Theorem, which suggests that any rational root (if it exists) would be a factor of the constant term (-4) divided by a factor of the leading coefficient (1). This gives possible rational roots of $\pm 1, \pm 2, \pm 4$.

2. Synthetic Division or Substitution: We can substitute each of these possible values to see if they yield zero (indicating a root). If a root is found, we could then perform synthetic division to reduce the polynomial to a cubic or quadratic, making it easier to solve.

3. Numerical or Graphical Methods: If none of the rational roots work, we may need to resort to numerical methods or graphing to approximate the roots of this polynomial.

4. Factoring by Grouping or Special Techniques: Fourth-degree polynomials sometimes allow factoring techniques specific to quartics, though they are often complex.

Would you like me to go through the Rational Root Theorem to check for any rational roots or approximate the solutions?

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Additional Related Questions

1. What are the possible rational roots of other polynomials using the Rational Root Theorem?
2. How can we approximate roots if rational roots are not found?
3. What specific techniques exist for factoring higher-degree polynomials?
4. What are the key differences between solving even- and odd-degree polynomials?
5. How does graphing help in approximating or visualizing the roots of polynomials?

Tip: When dealing with higher-degree polynomials, always start by checking for rational roots before attempting more complex methods.