Answer :

We want to factor completely the expression

[tex]$$
x^6y^2 + 2x^4y^2 - x^4y.
$$[/tex]

Step 1. Identify the Common Factor

First, we look for a common factor in all three terms. Notice that every term has at least a factor of [tex]$x^4$[/tex] and a factor of [tex]$y$[/tex]. So, we factor out [tex]$x^4y$[/tex]:

[tex]$$
x^6y^2 + 2x^4y^2 - x^4y = x^4y\left(\frac{x^6y^2}{x^4y} + \frac{2x^4y^2}{x^4y} - \frac{x^4y}{x^4y}\right).
$$[/tex]

Step 2. Simplify Each Term Inside the Parentheses

Divide each term by the common factor [tex]$x^4y$[/tex]:

1. For the first term:
[tex]$$
\frac{x^6y^2}{x^4y} = x^{6-4}y^{2-1} = x^2y.
$$[/tex]

2. For the second term:
[tex]$$
\frac{2x^4y^2}{x^4y} = 2y^{2-1} = 2y.
$$[/tex]

3. For the third term:
[tex]$$
\frac{x^4y}{x^4y} = 1.
$$[/tex]

So the expression inside the parentheses becomes:
[tex]$$
x^2y + 2y - 1.
$$[/tex]

Step 3. Write the Final Answer

Now, putting it all together, the completely factored form is:
[tex]$$
x^4y\left(x^2y + 2y - 1\right).
$$[/tex]

Thus, the completely factored form of the given expression is

[tex]$$
\boxed{x^4y\left(x^2y + 2y - 1\right)}.
$$[/tex]

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Rewritten by : Barada

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