$$ \left. \begin{array} { l } { C = ( \frac { 4 } { 3 } x - 1 ) ( 4 x + 1 ) + ( 2 x - \frac { 1 } { 4 } ) ^ { 2 } } \ { D = ( \frac { 3 x - 2 } { 6 } ) ^ { 2 } - ( \frac { 2 x + 1 } { 2 } ) ^ { 2 } } \ { E = ( \frac { 2 } { 5 } x + 1 ) ^ { 2 } - \frac { 4 } { 25 } ( 10 + \frac { x } { 2 } ) ^ { 2 } + \frac { x } { 10 } ( 7 + 4 x ) } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { C = ( \frac { 4 } { 3 } x - 1 ) ( 4 x + 1 ) + ( 2 x - \frac { 1 } { 4 } ) ^ { 2 } } \ { D = ( \frac { 3 x - 2 } { 6 } ) ^ { 2 } - ( \frac { 2 x + 1 } { 2 } ) ^ { 2 } } \ { E = ( \frac { 2 } { 5 } x + 1 ) ^ { 2 } - \frac { 4 } { 25 } ( 10 + \frac { x } { 2 } ) ^ { 2 } + \frac { x } { 10 } ( 7 + 4 x ) } \end{array} ight. $$

UpStudy AI · Accepted Answer

To analyze the expressions $$ C $$, $$ D $$, and $$ E $$, we will simplify each one step by step.

### Expression $$ C $$

$$
C = \left( \frac{4}{3} x - 1 \right)(4x + 1) + \left( 2x - \frac{1}{4} \right)^2
$$

1. **Expand the first term**:
   $$
   \left( \frac{4}{3} x - 1 \right)(4x + 1) = \frac{4}{3} x \cdot 4x + \frac{4}{3} x \cdot 1 - 1 \cdot 4x - 1 \cdot 1
   $$
   $$
   = \frac{16}{3} x^2 + \frac{4}{3} x - 4x - 1
   $$
   $$
   = \frac{16}{3} x^2 + \left( \frac{4}{3} - \frac{12}{3} \right)x - 1
   $$
   $$
   = \frac{16}{3} x^2 - \frac{8}{3} x - 1
   $$

2. **Expand the second term**:
   $$
   \left( 2x - \frac{1}{4} \right)^2 = (2x)^2 - 2 \cdot 2x \cdot \frac{1}{4} + \left( \frac{1}{4} \right)^2
   $$
   $$
   = 4x^2 - \frac{4x}{4} + \frac{1}{16} = 4x^2 - x + \frac{1}{16}
   $$

3. **Combine both parts**:
   $$
   C = \left( \frac{16}{3} x^2 - \frac{8}{3} x - 1 \right) + \left( 4x^2 - x + \frac{1}{16} \right)
   $$
   To combine, we need a common denominator for the constants:
   $$
   -1 = -\frac{16}{16}, \quad \text{and} \quad -\frac{8}{3} = -\frac{42.67}{16}
   $$
   Thus,
   $$
   C = \left( \frac{16}{3} + 4 \right)x^2 + \left( -\frac{8}{3} - 1 \right)x + \left( -1 + \frac{1}{16} \right)
   $$

### Expression $$ D $$

$$
D = \left( \frac{3x - 2}{6} \right)^2 - \left( \frac{2x + 1}{2} \right)^2
$$

1. **Expand both squares**:
   $$
   \left( \frac{3x - 2}{6} \right)^2 = \frac{(3x - 2)^2}{36} = \frac{9x^2 - 12x + 4}{36}
   $$
   $$
   \left( \frac{2x + 1}{2} \right)^2 = \frac{(2x + 1)^2}{4} = \frac{4x^2 + 4x + 1}{4}
   $$

2. **Combine**:
   $$
   D = \frac{9x^2 - 12x + 4}{36} - \frac{36x^2 + 36x + 9}{36}
   $$
   $$
   = \frac{(9x^2 - 12x + 4) - (36x^2 + 36x + 9)}{36}
   $$
   $$
   = \frac{-27x^2 - 48x - 5}{36}
   $$

### Expression $$ E $$

$$
E = \left( \frac{2}{5} x + 1 \right)^2 - \frac{4}{25} \left( 10 + \frac{x}{2} \right)^2 + \frac{x}{10} (7 + 4x)
$$

1. **Expand the first term**:
   \[
   \left( \frac{2}{5} x + 1 \right)^2 = \frac{4}{25} x^2 + \frac{4}{5} x + 
C = \frac{16}{3} x^2 - \frac{8}{3} x - 1 + 4x^2 - x + \frac{1}{16} D = \frac{-27x^2 - 48x - 5}{36} E = \frac{4}{25} x^2 + \frac{4}{5} x + 1 - \frac{4}{25} \left( 10 + \frac{x}{2} \right)^2 + \frac{x}{10} (7 + 4x)

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