The problem in the image reads:

"If $y = 1 - 2x - 2x^2 = 4x^4 - 3x^3$, then $x$ is:"

It is a multiple-choice question with options:

1. $3$
2. $1$
3. $0$
4. $-1$

Let's break it down and solve it step by step.

The equation given is: $y = 1 - 2x - 2x^2 = 4x^4 - 3x^3$

We can set the two sides equal to each other, because both expressions represent $y$: $1 - 2x - 2x^2 = 4x^4 - 3x^3$

Now, rearranging this into a standard form of a polynomial equation: $4x^4 - 3x^3 + 2x^2 + 2x - 1 = 0$

Next, we can try to solve this polynomial equation for $x$.The solutions to the equation $4x^4 - 3x^3 + 2x^2 + 2x - 1 = 0$ include both real and complex roots:

1. $x = \frac{-1 + \sqrt{17}}{8}$ (real root)
2. $x = \frac{1}{2} - \frac{\sqrt{3}}{2}i$ (complex root)
3. $x = \frac{1}{2} + \frac{\sqrt{3}}{2}i$ (complex root)
4. $x = \frac{-\sqrt{17}}{8} - \frac{1}{8}$ (real root)

Since the question is asking for a specific real solution from the given choices ($3$, $1$, $0$, and $-1$), none of these exact roots match the provided options.

However, simplifying or approximating the real roots shows that they are relatively small. Therefore, among the choices provided, $x = 0$ is the closest approximate solution.

Would you like further clarification or a more detailed explanation?

Here are some related questions to explore:

1. What techniques can be used to factor or approximate higher-degree polynomials?
2. How do you interpret complex solutions in polynomial equations?
3. What are the properties of real vs. complex roots in a polynomial?
4. Can substitution be used to simplify solving polynomial equations?
5. How can graphing help identify approximate solutions for polynomials?

Tip: Always check whether an equation has both real and complex solutions when solving polynomials of higher degrees!