https://file.aiquickdraw.com/mathbot/file/1725467879305bnc7xvx0.jpg

The equation from the uploaded image is:

\[
x^4 - 2x^3 - 20x^2 + 19x + 18 = 0
\]

And the provided solutions are: \( x = -2, -1.5, 3, 1 \).

Let's verify these solutions by substituting them into the equation.The provided solutions \((-2, -1.5, 3, 1)\) are not correct for the equation \(x^4 - 2x^3 - 20x^2 + 19x + 18 = 0\). When substituted back into the polynomial, they do not satisfy the equation (i.e., they do not result in 0):

- \(x = -2\) results in \(-68\)
- \(x = -1.5\) results in \(-43.6875\)
- \(x = 3\) results in \(-78\)
- \(x = 1\) results in \(16\)

None of these values satisfy the equation, indicating they are not the correct roots. If you need the correct roots, I can find them using a more precise method.

Would you like me to calculate the correct roots?

### Related Questions:
1. What is the general process to solve a fourth-degree polynomial?
2. How can we factorize polynomials to find their roots?
3. What numerical methods can be used to find approximate roots of polynomials?
4. How can we verify if a given solution is a root of an equation?
5. What are the differences between real and complex roots?

### Tip:
Always check if a proposed solution satisfies the original equation by substituting it back in.

Solve the polynomial equation given as x^4 - 2x^3 - 20x^2 + 19x + 18 = 0.

This problem involves solving a fourth-degree polynomial equation, x^4 - 2x^3 - 20x^2 + 19x + 18 = 0. The solution process includes verifying the proposed roots and identifying the correct roots using algebraic methods. Ideal for students in Grades 10-12.

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