Take away `((8)/(5)x^(2) - (2)/(3)x^(3) + (3)/(2)x -1)` from `((x^(3))/(5) - (3)/(2)x^(2) + (2)/(3)x + (1)/(4))`

Answer

To solve the problem of taking away the expression \(\frac{8}{5}x^2 - \frac{2}{3}x^3 + \frac{3}{2}x - 1\) from \(\frac{x^3}{5} - \frac{3}{2}x^2 + \frac{2}{3}x + \frac{1}{4}\), we will perform the subtraction step by step.

### Step-by-Step Solution:

1. **Write the expressions clearly**:
We have two expressions:
- Expression 1: \(\frac{x^3}{5} - \frac{3}{2}x^2 + \frac{2}{3}x + \frac{1}{4}\)
- Expression 2: \(\frac{8}{5}x^2 - \frac{2}{3}x^3 + \frac{3}{2}x - 1\)

We need to subtract Expression 2 from Expression 1.

2. **Set up the subtraction**:
\[ \left(\frac{x^3}{5} - \frac{3}{2}x^2 + \frac{2}{3}x + \frac{1}{4}\right) - \left(\frac{8}{5}x^2 - \frac{2}{3}x^3 + \frac{3}{2}x - 1\right) \]

3. **Distribute the negative sign**:
When we subtract the second expression, we need to change the signs of each term in it:
\[ \frac{x^3}{5} - \frac{3}{2}x^2 + \frac{2}{3}x + \frac{1}{4} - \frac{8}{5}x^2 + \frac{2}{3}x^3 - \frac{3}{2}x + 1 \]

4. **Combine like terms**:
Now, we will combine like terms:
- For \(x^3\): \(\frac{x^3}{5} + \frac{2}{3}x^3\)
- For \(x^2\): \(-\frac{3}{2}x^2 - \frac{8}{5}x^2\)
- For \(x\): \(\frac{2}{3}x - \frac{3}{2}x\)
- Constant terms: \(\frac{1}{4} + 1\)

5. **Finding a common denominator**:
- For \(x^3\): The common denominator of 5 and 3 is 15.
\[ \frac{x^3}{5} = \frac{3x^3}{15}, \quad \frac{2}{3}x^3 = \frac{10x^3}{15} \quad \Rightarrow \quad \frac{3x^3 + 10x^3}{15} = \frac{13x^3}{15} \]
- For \(x^2\): The common denominator of 2 and 5 is 10.
\[ -\frac{3}{2}x^2 = -\frac{15}{10}x^2, \quad -\frac{8}{5}x^2 = -\frac{16}{10}x^2 \quad \Rightarrow \quad -\frac{15x^2 + 16x^2}{10} = -\frac{31x^2}{10} \]
- For \(x\): The common denominator of 3 and 2 is 6.
\[ \frac{2}{3}x = \frac{4}{6}x, \quad -\frac{3}{2}x = -\frac{9}{6}x \quad \Rightarrow \quad \frac{4x - 9x}{6} = -\frac{5x}{6} \]
- For constant terms: The common denominator of 4 and 1 is 4.
\[ 1 = \frac{4}{4} \quad \Rightarrow \quad \frac{1}{4} + \frac{4}{4} = \frac{5}{4} \]

6. **Final expression**:
Putting it all together, we have:
\[ \frac{13}{15}x^3 - \frac{31}{10}x^2 - \frac{5}{6}x + \frac{5}{4} \]

### Final Answer:
\[ \frac{13}{15}x^3 - \frac{31}{10}x^2 - \frac{5}{6}x + \frac{5}{4} \]