Answer :

Let's simplify the expression [tex]\(48x^2 - 12x^3 - 24x^4\)[/tex] step by step.

1. Identify the Common Factor:
We first look for the greatest common factor in each term of the expression. The terms are [tex]\(48x^2\)[/tex], [tex]\(-12x^3\)[/tex], and [tex]\(-24x^4\)[/tex].

- The coefficients are 48, -12, and -24. The greatest common factor of these coefficients is 12.
- For the variable part, we notice the smallest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].

Hence, the greatest common factor for the entire expression is [tex]\(12x^2\)[/tex].

2. Factor out the Common Factor:
We factor out [tex]\(12x^2\)[/tex] from each term:

[tex]\[
48x^2 - 12x^3 - 24x^4 = 12x^2 \left(\frac{48x^2}{12x^2}\right) - 12x^2 \left(\frac{-12x^3}{12x^2}\right) - 12x^2 \left(\frac{-24x^4}{12x^2}\right)
\][/tex]

Simplifying inside the parentheses:

[tex]\[
48x^2 - 12x^3 - 24x^4 = 12x^2 \left(4\right) - 12x^2 \left(-x\right) - 12x^2 \left(2x^2\right)
\][/tex]

3. Write the Simplified Expression:

[tex]\[
48x^2 - 12x^3 - 24x^4 = 12x^2 \left(4 - x - 2x^2\right)
\][/tex]

4. Rearrange the Terms:
Generally, we write polynomial terms in descending order of their degrees. So we rearrange the terms inside the parentheses:

[tex]\[
48x^2 - 12x^3 - 24x^4 = 12x^2 \left(-2x^2 - x + 4\right)
\][/tex]

Thus, the simplified form of the expression [tex]\(48x^2 - 12x^3 - 24x^4\)[/tex] is:

[tex]\[
12x^2(-2x^2 - x + 4)
\][/tex]

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