1. **Problem Statement:** Divide the polynomial $P(x) = 3x^2 + 5x - 4$ by $D(x) = x + 3$ using synthetic or long division, and express $P(x)$ in the form $$P(x) = D(x) \cdot Q(x) + R(x).$$ 2. **Formula and Rules:** Polynomial division is similar to numerical division. We find a quotient polynomial $Q(x)$ and a remainder polynomial $R(x)$ such that the degree of $R(x)$ is less than the degree of $D(x)$. Here, $D(x)$ is linear, so $R(x)$ will be a constant. 3. **Set up synthetic division:** Since $D(x) = x + 3$, the root is $-3$. Write coefficients of $P(x)$: 3 (for $x^2$), 5 (for $x$), and -4 (constant). 4. **Perform synthetic division:** - Bring down 3. - Multiply 3 by -3: $3 \times (-3) = -9$. - Add to 5: $5 + (-9) = -4$. - Multiply -4 by -3: $-4 \times (-3) = 12$. - Add to -4: $-4 + 12 = 8$. 5. **Interpret results:** Quotient coefficients are 3 and -4, so $$Q(x) = 3x - 4,$$ and remainder is 8. 6. **Write final expression:** $$P(x) = (x + 3)(3x - 4) + 8.$$