To solve or factor the polynomial \( 2x^4 + 4x^3 - 24x^2 + 12x - 90 \), we'll first look for possible factorizations.

### Step 1: Factor out the greatest common factor (GCF)
First, notice that each term is divisible by 2, so we can factor out the 2:

\[
2(x^4 + 2x^3 - 12x^2 + 6x - 45)
\]

### Step 2: Look for possible factorization of the polynomial
Next, we'll attempt to factor the quartic polynomial \( x^4 + 2x^3 - 12x^2 + 6x - 45 \). This isn't easily factorable by simple methods, so let's consider applying synthetic division or checking for rational roots.

Let's test the rational roots using the Rational Root Theorem, which suggests possible roots are factors of the constant term (±1, ±3, ±5, ±9, ±15, ±45) divided by the leading coefficient (±1). 

After testing, we find that \( x = 3 \) is a root.

### Step 3: Perform synthetic division
Now, we perform synthetic division of \( x^4 + 2x^3 - 12x^2 + 6x - 45 \) by \( x - 3 \).

Dividing \( x^4 + 2x^3 - 12x^2 + 6x - 45 \) by \( x - 3 \) gives:

\[
(x^4 + 2x^3 - 12x^2 + 6x - 45) \div (x - 3) = x^3 + 5x^2 + 3x - 15
\]

### Step 4: Factor the cubic polynomial
Now, we have:

\[
2(x - 3)(x^3 + 5x^2 + 3x - 15)
\]

We can further factor \( x^3 + 5x^2 + 3x - 15 \) by checking for rational roots again. \( x = 3 \) is again a root. Dividing \( x^3 + 5x^2 + 3x - 15 \) by \( x - 3 \) gives:

\[
x^3 + 5x^2 + 3x - 15 = (x - 3)(x^2 + 8x + 5)
\]

### Step 5: Combine all factors
Now, we combine everything:

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

This is the factored form of the original polynomial.

### Summary
The polynomial \( 2x^4 + 4x^3 - 24x^2 + 12x - 90 \) factors as:

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

Would you like more details or have any questions?

Here are some related questions:
1. How do you perform synthetic division?
2. What is the Rational Root Theorem, and how is it used?
3. How do you factor cubic polynomials?
4. What are the applications of factoring polynomials in real-world problems?
5. How can you verify the roots of a polynomial equation?

**Tip:** When factoring polynomials, always start by factoring out the greatest common factor before attempting more complex methods.

Learn how to factorize the polynomial 2x^4 + 4x^3 - 24x^2 + 12x - 90 step-by-step. This problem involves polynomial factorization techniques, synthetic division, and application of the Rational Root Theorem. Suitable for high school students in Grades 10-12.

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