Sure! Let's find the roots of the polynomial \( 2x^4 + 4x^3 - 24x^2 + 12x - 90 \) using the factored form we derived earlier:

\[
2(x - 3)^2(x^2 + 8x + 5) = 0
\]

### Step 1: Solve \( (x - 3)^2 = 0 \)
For \( (x - 3)^2 = 0 \), the solution is:

\[
x = 3
\]

Since it's a squared factor, \( x = 3 \) is a root with multiplicity 2.

### Step 2: Solve \( x^2 + 8x + 5 = 0 \)
Now, let's find the roots of the quadratic equation \( x^2 + 8x + 5 = 0 \) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 1 \), \( b = 8 \), and \( c = 5 \). Plugging in these values:

\[
x = \frac{-8 \pm \sqrt{8^2 - 4(1)(5)}}{2(1)} = \frac{-8 \pm \sqrt{64 - 20}}{2} = \frac{-8 \pm \sqrt{44}}{2} = \frac{-8 \pm \sqrt{4 \times 11}}{2} = \frac{-8 \pm 2\sqrt{11}}{2} 
\]

Simplifying:

\[
x = -4 \pm \sqrt{11}
\]

So, the roots from the quadratic equation are:

\[
x = -4 + \sqrt{11} \quad \text{and} \quad x = -4 - \sqrt{11}
\]

### Final List of Roots
The roots of the polynomial \( 2x^4 + 4x^3 - 24x^2 + 12x - 90 \) are:

1. \( x = 3 \) (with multiplicity 2)
2. \( x = -4 + \sqrt{11} \)
3. \( x = -4 - \sqrt{11} \)

Would you like more details or have any questions?

Here are some related questions:
1. What does the multiplicity of a root mean?
2. How do you use the quadratic formula?
3. How can you graph a polynomial using its roots?
4. What is the significance of the discriminant in the quadratic formula?
5. How do complex roots arise in polynomials?

**Tip:** When finding roots, always check if any roots have multiplicity greater than 1, as they affect the shape of the polynomial's graph.

Learn how to find the roots of the polynomial 2x^4 + 4x^3 - 24x^2 + 12x - 90 with a detailed step-by-step solution. This problem covers key algebraic concepts, including polynomials, quadratic equations, and the application of the quadratic formula. Suitable for students in Grades 10-12.

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