\( 1,12.11 \frac { x + 2 } { 4 x ^ { 3 } - 2 x + 1 } - \frac { 8 x + 1 } { 8 x ^ { 3 } + 1 } \)

Simplify the expression by following steps: - step0: Solution: \(1.12+\frac{11\left(x+2\right)}{\left(4x^{3}-2x+1\right)}-\frac{\left(8x+1\right)}{\left(8x^{3}+1\right)}\) - step1: Remove the parentheses: \(1.12+\frac{11\left(x+2\right)}{4x^{3}-2x+1}-\frac{8x+1}{8x^{3}+1}\) - step2: Rewrite the expression: \(\frac{28}{25}+\frac{11\left(x+2\right)}{4x^{3}-2x+1}-\frac{8x+1}{8x^{3}+1}\) - step3: Reduce fractions to a common denominator: \(\frac{28\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}{25\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}+\frac{11\left(x+2\right)\times 25\left(8x^{3}+1\right)}{\left(4x^{3}-2x+1\right)\times 25\left(8x^{3}+1\right)}-\frac{\left(8x+1\right)\times 25\left(4x^{3}-2x+1\right)}{\left(8x^{3}+1\right)\times 25\left(4x^{3}-2x+1\right)}\) - step4: Reorder the terms: \(\frac{28\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}{25\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}+\frac{11\left(x+2\right)\times 25\left(8x^{3}+1\right)}{25\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}-\frac{\left(8x+1\right)\times 25\left(4x^{3}-2x+1\right)}{\left(8x^{3}+1\right)\times 25\left(4x^{3}-2x+1\right)}\) - step5: Reorder the terms: \(\frac{28\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}{25\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}+\frac{11\left(x+2\right)\times 25\left(8x^{3}+1\right)}{25\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}-\frac{\left(8x+1\right)\times 25\left(4x^{3}-2x+1\right)}{25\left(8x^{3}+1\right)\left(4x^{3}-2x+1\right)}\) - step6: Rewrite the expression: \(\frac{28\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}{25\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}+\frac{11\left(x+2\right)\times 25\left(8x^{3}+1\right)}{25\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}-\frac{\left(8x+1\right)\times 25\left(4x^{3}-2x+1\right)}{25\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}\) - step7: Transform the expression: \(\frac{28\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)+11\left(x+2\right)\times 25\left(8x^{3}+1\right)-\left(8x+1\right)\times 25\left(4x^{3}-2x+1\right)}{25\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}\) - step8: Multiply the terms: \(\frac{896x^{6}+336x^{3}-448x^{4}-56x+28+11\left(x+2\right)\times 25\left(8x^{3}+1\right)-\left(8x+1\right)\times 25\left(4x^{3}-2x+1\right)}{25\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}\) - step9: Multiply the terms: \(\frac{896x^{6}+336x^{3}-448x^{4}-56x+28+2200x^{4}+275x+4400x^{3}+550-\left(8x+1\right)\times 25\left(4x^{3}-2x+1\right)}{25\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}\) - step10: Multiply the terms: \(\frac{896x^{6}+336x^{3}-448x^{4}-56x+28+2200x^{4}+275x+4400x^{3}+550-\left(800x^{4}-400x^{2}+150x+100x^{3}+25\right)}{25\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}\) - step11: Calculate: \(\frac{896x^{6}+4636x^{3}+952x^{4}+69x+553+400x^{2}}{25\left(4x^{3}-2x+1\right)\left(8x^{3}+1\right)}\) - step12: Calculate: \(\frac{896x^{6}+4636x^{3}+952x^{4}+69x+553+400x^{2}}{800x^{6}+300x^{3}-400x^{4}-50x+25}\) To simplify the expression \[ 1.12 + 11 \cdot \frac{x + 2}{4x^3 - 2x + 1} - \frac{8x + 1}{8x^3 + 1}, \] we can break it down into steps: 1. **Convert the decimal to a fraction**: \(1.12\) can be expressed as \(\frac{112}{100} = \frac{28}{25}\). 2. **Combine the fractions**: The expression now looks like: \[ \frac{28}{25} + 11 \cdot \frac{x + 2}{4x^3 - 2x + 1} - \frac{8x + 1}{8x^3 + 1}. \] 3. **Find a common denominator**: The common denominator for the fractions will be the product of the denominators: \[ (4x^3 - 2x + 1)(8x^3 + 1) \cdot 25. \] 4. **Combine all terms over the common denominator**: After performing the necessary multiplications and combining, we arrive at the simplified expression: \[ \frac{896x^{6}+4636x^{3}+952x^{4}+69x+553+400x^{2}}{800x^{6}+300x^{3}-400x^{4}-50x+25}. \] Thus, the final simplified form of the expression is: \[ \frac{896x^{6}+4636x^{3}+952x^{4}+69x+553+400x^{2}}{800x^{6}+300x^{3}-400x^{4}-50x+25}. \]