2x⁴-16x²+32

To analyze the expression \( 2x^4 - 16x^2 + 32 \), we can start by factoring it.

First, let's factor out the common term:

\[ 2(x^4 - 8x^2 + 16) \]

Now, we can simplify the quadratic part \( x^4 - 8x^2 + 16 \) by letting \( y = x^2 \), which transforms our expression into a quadratic form:

\[ y^2 - 8y + 16 \]

Next, we can factor the quadratic:

\[ (y - 4)^2 \]

Substituting back \( y = x^2 \), we have:

\[ (x^2 - 4)^2 \]

Now, we can further factor \( x^2 - 4 \) as it is a difference of squares:

\[ (x - 2)(x + 2) \]

Therefore, we substitute back to get the complete factorization of the original expression:

\[ 2[(x - 2)(x + 2)]^2 \]

Putting it all together, the fully factored form of \( 2x^4 - 16x^2 + 32 \) is:

\[ 2(x - 2)^2(x + 2)^2 \]

This expression can be used in further calculations or to find roots if needed. If you have any specific questions or need further explanations, let me know!