1. **State the problem:** Find the roots of the polynomial equation $$4x^3 + 7x^2 - 5x - 6 = 0$$ using synthetic division. 2. **Recall the Rational Root Theorem:** Possible rational roots are factors of the constant term divided by factors of the leading coefficient. Here, constant term = -6, leading coefficient = 4. 3. **Possible roots:** $$\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{1}{4}, \pm \frac{3}{4}$$ 4. **Test roots using synthetic division:** Start with $x=1$. Synthetic division setup for $x=1$: Coefficients: 4 | 7 | -5 | -6 Bring down 4. Multiply 4*1=4, add to 7 = 11. Multiply 11*1=11, add to -5 = 6. Multiply 6*1=6, add to -6 = 0. Remainder is 0, so $x=1$ is a root. 5. **Divide polynomial by $(x-1)$:** Quotient polynomial is $$4x^2 + 11x + 6$$ 6. **Factor the quadratic:** $$4x^2 + 11x + 6 = (4x + 3)(x + 2)$$ 7. **Find roots of factors:** $$4x + 3 = 0 \Rightarrow x = -\frac{3}{4}$$ $$x + 2 = 0 \Rightarrow x = -2$$ 8. **Final roots:** $$x = 1, -\frac{3}{4}, -2$$ This completes the solution for the first polynomial.