Answer :

Let's factor the expression [tex]\(4x^4 - 32x^3 + 48x^2\)[/tex] completely step-by-step:

1. Identify the Greatest Common Factor (GCF):
Look at the coefficients and the powers of [tex]\(x\)[/tex] in each term. The terms are [tex]\(4x^4\)[/tex], [tex]\(-32x^3\)[/tex], and [tex]\(48x^2\)[/tex]. The GCF of the coefficients (4, 32, and 48) is 4. Each term is also divisible by [tex]\(x^2\)[/tex] since the smallest power among them is [tex]\(x^2\)[/tex].

2. Factor out the GCF:
We factor out [tex]\(4x^2\)[/tex] from each term:
[tex]\[
4x^2(x^2 - 8x + 12)
\][/tex]

3. Factor the quadratic expression inside the parentheses:
Now, we focus on factoring the quadratic [tex]\(x^2 - 8x + 12\)[/tex]:
- Look for two numbers that multiply to 12 (the constant term) and add up to -8 (the coefficient of [tex]\(x\)[/tex]).
- These numbers are -6 and -2.

4. Write the factors of the quadratic:
The expression [tex]\(x^2 - 8x + 12\)[/tex] can be factored as:
[tex]\[
(x - 6)(x - 2)
\][/tex]

5. Combine the factors:
Now substitute back into the expression:
[tex]\[
4x^2(x - 6)(x - 2)
\][/tex]

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

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