1. **State the problem:** Simplify the expression $$(4x^2 + x + 3)(x + 2)$$ and find the coefficients of the resulting cubic polynomial in the form $$ax^3 + bx^2 + cx + d$$. 2. **Formula and rules:** To multiply polynomials, use the distributive property (also called FOIL for binomials), multiplying each term in the first polynomial by each term in the second polynomial. 3. **Multiply each term:** - Multiply $4x^2$ by $x$ to get $$4x^3$$. - Multiply $4x^2$ by $2$ to get $$8x^2$$. - Multiply $x$ by $x$ to get $$x^2$$. - Multiply $x$ by $2$ to get $$2x$$. - Multiply $3$ by $x$ to get $$3x$$. - Multiply $3$ by $2$ to get $$6$$. 4. **Combine like terms:** $$8x^2 + x^2 = 9x^2$$ $$2x + 3x = 5x$$ 5. **Write the final expression:** $$4x^3 + 9x^2 + 5x + 6$$ 6. **Answer:** The coefficients are: - $a = 4$ - $b = 9$ - $c = 5$ - $d = 6$