Answer :

To perform the division [tex]\(\frac{-12 x^3 + 45 x^4}{9 x^4}\)[/tex], follow these steps:

1. Separate the Terms in the Numerator: Break down the expression in the numerator into separate fractions over the common denominator.

[tex]\[
\frac{-12 x^3 + 45 x^4}{9 x^4} = \frac{-12 x^3}{9 x^4} + \frac{45 x^4}{9 x^4}
\][/tex]

2. Simplify Each Fraction:

- For the first fraction [tex]\(\frac{-12 x^3}{9 x^4}\)[/tex]:
- Simplify the coefficients: [tex]\(\frac{-12}{9} = -\frac{4}{3}\)[/tex].
- Simplify the powers of [tex]\(x\)[/tex]: [tex]\(\frac{x^3}{x^4} = \frac{1}{x}\)[/tex].

So, [tex]\(\frac{-12 x^3}{9 x^4}\)[/tex] simplifies to [tex]\(-\frac{4}{3x}\)[/tex].

- For the second fraction [tex]\(\frac{45 x^4}{9 x^4}\)[/tex]:
- Simplify the coefficients: [tex]\(\frac{45}{9} = 5\)[/tex].
- Since the powers of [tex]\(x\)[/tex] are the same, [tex]\(\frac{x^4}{x^4} = 1\)[/tex].

So, [tex]\(\frac{45 x^4}{9 x^4} = 5\)[/tex].

3. Combine the Results: Combine the simplified fractions:

[tex]\[
\frac{-12 x^3}{9 x^4} + \frac{45 x^4}{9 x^4} = -\frac{4}{3x} + 5
\][/tex]

Therefore, the result of the division is:

[tex]\[
5 - \frac{4}{3x}
\][/tex]

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