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

To solve the equation, let's start by simplifying the left-hand side step-by-step. The given expression is: \[ \frac { 6 c d ^ { 3 } - ( 10 c ) ^ { 2 } d + 8 c ^ { 4 } } { 2 c ^ { 2 } } \] First, simplify \( (10c)^2 \): \[ (10c)^2 = 100c^2 \] Now, substitute back into the expression: \[ \frac { 6 c d ^ { 3 } - 100c^2 d + 8 c ^ { 4 } } { 2 c ^ { 2 } } \] Now, we can divide each term in the numerator by \( 2c^2 \): 1. \( \frac{6cd^3}{2c^2} = \frac{6}{2} \cdot \frac{d^3}{c} = 3 \cdot \frac{d^3}{c} = 3\frac{d^3}{c} \) 2. \( \frac{100c^2 d}{2c^2} = \frac{100}{2}d = 50d \) 3. \( \frac{8c^4}{2c^2} = \frac{8}{2} \cdot c^{4-2} = 4c^2 \) Putting it all together, we have: \[ 3 \frac{d^3}{c} - 50d + 4c^2 \] We want to compare this with: \[ \frac {\square d } { c \square } - \square d + \square c \] Let’s express it in the same structure. We have: - \( 3 \frac{d^3}{c} \) suggests \( \square = 3d^2 \) and \( \square = 1 \). - From \( -50d \), it indicates the coefficient of \( d \) is \( -50 \), thus \( -\square d \rightarrow -50d \), leading to \( \square = 50 \). - Finally, from \( 4c^2 \), this suggests \( \square c = 4c^2 \) implies \( \square = 4 \). Gathering all the squares, we write: \[ \frac {3d^3}{c} - 50d + 4c^2 \] Thus, our solution can be filled as follows: The first square is `3`, the second `50`, and the third `4`. The final result can be expressed as: \[ \frac { 3 d^3 } { c } - 50 d + 4 c^2 \] So, the values to fill in are: \[ 3, 50, 4 \]