Answer :

Sure! Let's go through the process of factoring the polynomial completely, step by step:

We are given the polynomial:

[tex]\[ 4x^4 - 32x^3 + 48x^2 \][/tex]

1. Find the Greatest Common Factor (GCF):
- Look at the coefficients: 4, -32, and 48. The GCF of these numbers is 4.
- Look at the variable parts: [tex]\( x^4, x^3, \)[/tex] and [tex]\( x^2 \)[/tex]. The smallest exponent is [tex]\( x^2 \)[/tex], so that's part of the GCF.
- Thus, the GCF of the entire expression is [tex]\( 4x^2 \)[/tex].

2. Factor out the GCF:
- We divide each term by the GCF, [tex]\( 4x^2 \)[/tex].
- This gives us:
[tex]\[
4x^2(x^2 - 8x + 12)
\][/tex]

3. Factor the quadratic expression:
- We need to factor [tex]\( x^2 - 8x + 12 \)[/tex].
- Look for two numbers that multiply to 12 and add to -8. These numbers are -6 and -2.
- So, [tex]\( x^2 - 8x + 12 \)[/tex] can be factored as [tex]\( (x - 6)(x - 2) \)[/tex].

4. Combine the factors:
- Putting it all together, the completely factored form of the polynomial is:
[tex]\[
4x^2(x - 6)(x - 2)
\][/tex]

So, the polynomial [tex]\( 4x^4 - 32x^3 + 48x^2 \)[/tex] factors completely to [tex]\( 4x^2(x - 6)(x - 2) \)[/tex].

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