What is the simplest form of the expression 6x^4 + 16x^2 + 9x + 1?

To simplify the expression 6x⁴ + 16x² + 9x + 1, we will look for common factors and see if we can factor it further.

First, let’s rewrite the expression clearly:

6x⁴ + 16x² + 9x + 1

Now, notice that this expression doesn’t have a straightforward factorization due to the different powers of x. We can consider arranging it in descending powers:

6x⁴ + 0x³ + 16x² + 9x + 1

Next, we can try to group the terms or use synthetic division or the rational root theorem, but let’s also check if it can be factored directly:

After attempting various techniques, we find that this polynomial does not factor neatly into simpler binomials. However, if you’re looking for specific values, you could evaluate it at various points or use polynomial long division if needed.

As a final note, unless there are specific constraints on the values of x or additional context provided that allows further simplification or substitution, the expression remains as:

6x⁴ + 16x² + 9x + 1

This expression can be evaluated or graphed as needed, but in its simplest algebraic form, it does not break down further easily.