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Answer:

[tex]\huge\boxed{x(a-x)(a+x)^2}[/tex]

Step-by-step explanation:

In order to factor this expression, our goal is to write the expression in a way that we can factor out a term.

With the expression [tex]a^3 x - x^4 + a^2 x^2 - ax^3[/tex], we need to note an exponent rule.

[tex]a^{b+c} = a^b a^c[/tex]

Step 1:

We can use this to get each term of this expression to have a term of [tex]x[/tex] so we can factor it out.

Let's look at each term and get it so that we can factor out an x term.

  • [tex]a^3 x = x a^3[/tex] (Commutative Property)
  • [tex]x^4 = xx^3[/tex] (Since [tex]x^3 \cdot x^1 = x^4[/tex])
  • [tex]a^2 x^2 = a^2 \cdot xx[/tex] (Since [tex]x^2 = x \cdot x[/tex])
  • [tex]ax^3 = a \cdot xx^2[/tex] (Since [tex]x^3 = x^2 \cdot x^1[/tex]

With this, our equation becomes [tex](xa^3) - (xx^3) + (xxa^2) - (xx^2a)[/tex].

We now can factor out the common term x.

[tex]x(a^3 - x^3 + xa^2 - x^2a)[/tex]

Step 2:

From here, we can now factor [tex]a^3 - x^3 + xa^2 - x^2 a[/tex]

  • Rearrange the equation: [tex]a^3 +xa^2 - x^3 - x^2a[/tex]
  • Factor out [tex]a^2[/tex] from [tex]a^3 + xa^2[/tex]  which comes out to be  [tex]a^2(a+x)[/tex]
  • Factor out [tex]-x^2[/tex] from [tex]-x^3 - x^2a[/tex] which comes out to be [tex]-x^2(x+a)[/tex]
  • We now have [tex]a^2(a+x) - x^2(x+a)[/tex]
  • Factor out the common term, [tex](a+x)[/tex], which comes out to be [tex](a+x)(a^2 - x^2)[/tex]
  • Factor [tex]-x^2+a^2[/tex] into [tex](a+x)(a-x)[/tex]
  • We now have [tex](a+x) (a+x) (a-x)[/tex], which is simplified to [tex](a-x)(a+x)^2[/tex]

Finalizing:

Since we have just factored [tex]a^3 - x^3 + xa^2 - x^2 a[/tex]  and factored x out of [tex]a^3 x - x^4 + a^2 x^2 - ax^3[/tex] in the first couple of steps, we need to have it as a factorization of x.

[tex]x(a-x)(a+x)^2[/tex]

Hope this helped!