Answer:

The equation is:

f(x) = (x^3 + 4x^2 + 2x - 1)/(2x^2 - 3x + 1) = 0

To simplify this expression, let's factor the numerator and denominator separately.

Factoring the numerator:

x^3 + 4x^2 + 2x - 1 = 0

We can use synthetic division or polynomial long division to find one of the roots. By observing the equation, we can see that x = 1 is a root.

Using synthetic division, we get:

1 | 1 4 2 -1

| 1 5 7

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1 5 7 6

The result of the synthetic division is: x^2 + 5x + 7 with a remainder of 6.

So, x^3 + 4x^2 + 2x - 1 can be factored as (x - 1)(x^2 + 5x + 7) + 6.

Factoring the denominator:

2x^2 - 3x + 1 = 0

This quadratic equation can be factored as (2x - 1)(x - 1).

Now, let's rewrite the equation f(x) = 0 using the factored expressions:

[(x - 1)(x^2 + 5x + 7) + 6] / [(2x - 1)(x - 1)] = 0

Now, we have three factors in the numerator: (x - 1), (x^2 + 5x + 7), and the constant 6. For f(x) to be zero, one or more of these factors must be zero.

Setting each factor equal to zero:

x - 1 = 0 => x = 1

x^2 + 5x + 7 = 0

We can apply the quadratic formula to solve for x:

x = (-b ± √(b^2 - 4ac)) / (2a)

Here, a = 1, b = 5, and c = 7.

x = (-5 ± √(5^2 - 4(1)(7))) / (2(1))

x = (-5 ± √(25 - 28)) / 2

x = (-5 ± √(-3)) / 2

Since we have a negative value under the square root, the quadratic equation x^2 + 5x + 7 = 0 does not have any real solutions. Hence, there are no additional values of x for which f(x) is equal to zero.

Therefore, the only value of x for which f(x) is equal to zero is x = 1.