Answer :

To find the quotient of the polynomial [tex]\(-35x^5 - 40x^4 + 15x^3 - 15x^2\)[/tex] when divided by [tex]\(-5x^2\)[/tex], we perform polynomial division. Here's a step-by-step explanation:

1. Identify the terms to divide:
- We have the polynomial [tex]\(-35x^5 - 40x^4 + 15x^3 - 15x^2\)[/tex] as the numerator.
- The denominator is [tex]\(-5x^2\)[/tex].

2. Divide each term of the polynomial by the denominator:
- For the first term: [tex]\(-35x^5\)[/tex] divided by [tex]\(-5x^2\)[/tex] gives [tex]\(7x^{5-2} = 7x^3\)[/tex].
- For the second term: [tex]\(-40x^4\)[/tex] divided by [tex]\(-5x^2\)[/tex] gives [tex]\(8x^{4-2} = 8x^2\)[/tex].
- For the third term: [tex]\(15x^3\)[/tex] divided by [tex]\(-5x^2\)[/tex] gives [tex]\(-3x^{3-2} = -3x\)[/tex].
- For the fourth term: [tex]\(-15x^2\)[/tex] divided by [tex]\(-5x^2\)[/tex] gives [tex]\(3x^{2-2} = 3\)[/tex].

3. Combine the results:
- The quotient of the division is [tex]\(7x^3 + 8x^2 - 3x + 3\)[/tex].

So, the quotient is [tex]\(7x^3 + 8x^2 - 3x + 3\)[/tex].

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