1. **State the problem:** We need to expand and simplify the expression $$(2x-1)(x^2+4x+7)$$ and write it in the form $$ax^3 + bx^2 + cx + d$$. Then find the value of $$b+c$$. 2. **Use the distributive property (FOIL for polynomials):** Multiply each term in the first polynomial by each term in the second polynomial: $$ (2x)(x^2) + (2x)(4x) + (2x)(7) - (1)(x^2) - (1)(4x) - (1)(7) $$ 3. **Calculate each product:** $$ 2x \cdot x^2 = 2x^3 $$ $$ 2x \cdot 4x = 8x^2 $$ $$ 2x \cdot 7 = 14x $$ $$ -1 \cdot x^2 = -x^2 $$ $$ -1 \cdot 4x = -4x $$ $$ -1 \cdot 7 = -7 $$ 4. **Combine like terms:** $$ 2x^3 + (8x^2 - x^2) + (14x - 4x) - 7 = 2x^3 + 7x^2 + 10x - 7 $$ 5. **Identify coefficients:** $$a = 2, \quad b = 7, \quad c = 10, \quad d = -7 $$ 6. **Find $$b+c$$:** $$b + c = 7 + 10 = 17$$ **Final answer:** $$b+c=17$$