Question

Im at a loss
i would really appreciate some help

Answer 1

Answer:
`-10d^4+17d^2s-6s^2` (Choice A)

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Step-by-step explanation

I'll go over two methods.

Method 1

`(-2d^2+s)(5d^2-6s)\\\\=c(5d^2-6s) \ \ \text{ .... let } c = -2d^2+s\\\\=5cd^2-6cs \ \ \ \text{ ... distribute}\\\\=5d^2(c) - 6s(c)\\\\=5d^2(-2d^2+s) - 6s(-2d^2+s)\ \ \ \text{ ... plug in } c = -2d^2+s\\\\=-10d^4+5d^2s + 12d^2s-6s^2\ \ \ \text{ ... distribute twice more}\\\\=-10d^4+17d^2s-6s^2\ \ \ \text{ ... combine like terms}\\\\`

Therefore,

`(-2d^2+s)(5d^2-6s)=-10d^4+17d^2s-6s^2`

is an identity that's true for all real numbers d and s.

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Method 2

For this approach, I'll use the box method. This is a visual way to multiply out polynomials.

The idea is that we place the terms of
`-2d^2+s` and
`5d^2-6s` along the top and left sides as shown in the diagram below.

The four red terms inside the boxes are the result of multiplying the outer terms. For example, the
`-10d^4` in the upper left corner is the result of multiplying
`-2d^2` and
`5d^2` together.

Once we determine what goes in those four inner boxes, we then add up the terms and simplify. That leads to the final answer we got earlier which was
`-10d^4+17d^2s-6s^2`