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Answer :
We start with the expression
[tex]$$48x^2 - 12x^3 - 24x^4.$$[/tex]
Step 1. Factor Out the Greatest Common Factor
Observe that each term has a factor of [tex]$12x^2$[/tex]. Factoring this common factor out, we get
[tex]$$48x^2 - 12x^3 - 24x^4 = 12x^2\left( \frac{48x^2}{12x^2} - \frac{12x^3}{12x^2} - \frac{24x^4}{12x^2} \right).$$[/tex]
Simplify each fraction:
[tex]\[
\frac{48x^2}{12x^2} = 4, \quad \frac{12x^3}{12x^2} = x, \quad \frac{24x^4}{12x^2} = 2x^2.
\][/tex]
Thus, we have:
[tex]$$48x^2 - 12x^3 - 24x^4 = 12x^2\left(4 - x - 2x^2\right).$$[/tex]
Step 2. Rewrite the Inner Expression
Notice that the expression inside the parentheses can be rearranged as
[tex]$$4 - x - 2x^2.$$[/tex]
Multiplying the entire expression by [tex]$-1$[/tex] (which is equivalent to factoring [tex]$-1$[/tex] out) gives an equivalent form:
[tex]$$4 - x - 2x^2 = -\Bigl(2x^2 + x - 4\Bigr).$$[/tex]
Substitute this back into the factored form:
[tex]$$48x^2 - 12x^3 - 24x^4 = 12x^2\Bigl[-\bigl(2x^2 + x - 4\bigr)\Bigr].$$[/tex]
Step 3. Write the Final Factored Form
Multiplying the constants, we have:
[tex]$$12x^2\Bigl[-\bigl(2x^2 + x - 4\bigr)\Bigr] = -12x^2\left(2x^2 + x - 4\right).$$[/tex]
Thus, the completely factored expression is:
[tex]$$\boxed{-12x^2\left(2x^2 + x - 4\right)}.$$[/tex]
[tex]$$48x^2 - 12x^3 - 24x^4.$$[/tex]
Step 1. Factor Out the Greatest Common Factor
Observe that each term has a factor of [tex]$12x^2$[/tex]. Factoring this common factor out, we get
[tex]$$48x^2 - 12x^3 - 24x^4 = 12x^2\left( \frac{48x^2}{12x^2} - \frac{12x^3}{12x^2} - \frac{24x^4}{12x^2} \right).$$[/tex]
Simplify each fraction:
[tex]\[
\frac{48x^2}{12x^2} = 4, \quad \frac{12x^3}{12x^2} = x, \quad \frac{24x^4}{12x^2} = 2x^2.
\][/tex]
Thus, we have:
[tex]$$48x^2 - 12x^3 - 24x^4 = 12x^2\left(4 - x - 2x^2\right).$$[/tex]
Step 2. Rewrite the Inner Expression
Notice that the expression inside the parentheses can be rearranged as
[tex]$$4 - x - 2x^2.$$[/tex]
Multiplying the entire expression by [tex]$-1$[/tex] (which is equivalent to factoring [tex]$-1$[/tex] out) gives an equivalent form:
[tex]$$4 - x - 2x^2 = -\Bigl(2x^2 + x - 4\Bigr).$$[/tex]
Substitute this back into the factored form:
[tex]$$48x^2 - 12x^3 - 24x^4 = 12x^2\Bigl[-\bigl(2x^2 + x - 4\bigr)\Bigr].$$[/tex]
Step 3. Write the Final Factored Form
Multiplying the constants, we have:
[tex]$$12x^2\Bigl[-\bigl(2x^2 + x - 4\bigr)\Bigr] = -12x^2\left(2x^2 + x - 4\right).$$[/tex]
Thus, the completely factored expression is:
[tex]$$\boxed{-12x^2\left(2x^2 + x - 4\right)}.$$[/tex]
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