$$ \frac { 6 c d ^ { 3 } - ( 10 c ) ^ { 2 } d + 8 c ^ { 4 } } { 2 c ^ { 2 } } = \frac { \square d } { c \square } - \square d + \square c $$

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UpStudy AI · Accepted Answer

We start with the given expression

$$
\frac{6c\,d^3 - (10c)^2d + 8c^4}{2c^2}\,.
$$

Step 1. Expand the squared term in the numerator. Notice that

$$
(10c)^2 = 100c^2\,.
$$

Thus, the numerator becomes

$$
6c\,d^3 - 100c^2d + 8c^4\,.
$$

Step 2. Write the expression as three separate fractions:

$$
\frac{6c\,d^3}{2c^2} - \frac{100c^2d}{2c^2} + \frac{8c^4}{2c^2}\,.
$$

Step 3. Simplify each fraction term‐by‐term.

For the first term:

$$
\frac{6c\,d^3}{2c^2} = \frac{6d^3}{2c} = \frac{3d^3}{c}\,.
$$

For the second term:

$$
\frac{100c^2d}{2c^2} = \frac{100d}{2} = 50d\,.
$$

For the third term:

$$
\frac{8c^4}{2c^2} = \frac{8c^2}{2} = 4c^2\,.
$$

Step 4. Combine the simplified terms:

$$
\frac{3d^3}{c} - 50d + 4c^2\,.
$$

The original problem asks for an expression in the form

$$
\frac{\square d}{c\square} - \square d + \square c\,.
$$

By comparing, we identify:

- The first term is $$\frac{3d^3}{c}$$. (Here, the missing coefficient is $$3$$ and note that the power of $$d$$ is 3.)
- The second term is $$-50d$$. (So the missing number is $$50$$.)
- The third term is $$4c^2$$. (So the missing number is $$4$$ and the power of $$c$$ is 2.)

Thus, the filled expression is

$$
\frac{3d^3}{c} - 50d + 4c^2\,.
$$
$$ \frac{3d^3}{c} - 50d + 4c^2 $$

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