1. **Stating the problem:** Simplify the expression $$\frac{y(alla 3)+\frac{7}{4}y+1}{y+\frac{1}{2}}$$. It seems "alla 3" means $y^3$, so the expression is $$\frac{y^3 + \frac{7}{4}y + 1}{y + \frac{1}{2}}$$. 2. **Rewrite the expression clearly:** $$\frac{y^3 + \frac{7}{4}y + 1}{y + \frac{1}{2}}$$ 3. **Find a common denominator in numerator terms if needed:** Rewrite numerator as $$y^3 + \frac{7}{4}y + 1 = y^3 + \frac{7}{4}y + \frac{4}{4}$$ 4. **Try polynomial division or factorization:** We want to divide numerator by denominator: $$\frac{y^3 + \frac{7}{4}y + 1}{y + \frac{1}{2}}$$ 5. **Multiply numerator and denominator by 4 to clear fractions:** $$\frac{4y^3 + 7y + 4}{4y + 2}$$ 6. **Perform polynomial division:** Divide $4y^3 + 7y + 4$ by $4y + 2$. - First term: $\frac{4y^3}{4y} = y^2$ - Multiply divisor by $y^2$: $4y^3 + 2y^2$ - Subtract: $(4y^3 + 7y + 4) - (4y^3 + 2y^2) = -2y^2 + 7y + 4$ - Next term: $\frac{-2y^2}{4y} = -\frac{1}{2}y$ - Multiply divisor by $-\frac{1}{2}y$: $-2y^2 - y$ - Subtract: $(-2y^2 + 7y + 4) - (-2y^2 - y) = 8y + 4$ - Next term: $\frac{8y}{4y} = 2$ - Multiply divisor by $2$: $8y + 4$ - Subtract: $(8y + 4) - (8y + 4) = 0$ 7. **Result of division:** $$y^2 - \frac{1}{2}y + 2$$ 8. **Final answer:** $$\boxed{y^2 - \frac{1}{2}y + 2}$$