1. **State the problem:** We are given the trapezoid area formula $$A = \frac{(B + b) \cdot h}{2}$$ and expressions for $$A$$, $$b$$, and $$h$$ in terms of $$x$$. We need to find the expression for $$B$$. 2. **Write down the given values:** $$A = 12x^2 - 29x + 14$$ $$b = x - 5$$ $$h = 4x - 7$$ 3. **Substitute into the area formula:** $$12x^2 - 29x + 14 = \frac{(B + (x - 5)) \cdot (4x - 7)}{2}$$ 4. **Multiply both sides by 2 to clear the denominator:** $$2(12x^2 - 29x + 14) = (B + x - 5)(4x - 7)$$ $$24x^2 - 58x + 28 = (B + x - 5)(4x - 7)$$ 5. **Expand the right side:** $$(B + x - 5)(4x - 7) = B(4x - 7) + (x - 5)(4x - 7)$$ 6. **Expand $(x - 5)(4x - 7)$:** $$x \cdot 4x = 4x^2$$ $$x \cdot (-7) = -7x$$ $$-5 \cdot 4x = -20x$$ $$-5 \cdot (-7) = 35$$ Sum these: $$4x^2 - 7x - 20x + 35 = 4x^2 - 27x + 35$$ 7. **Rewrite the equation:** $$24x^2 - 58x + 28 = 4xB - 7B + 4x^2 - 27x + 35$$ 8. **Group terms and isolate terms with $$B$$:** $$24x^2 - 58x + 28 - 4x^2 + 27x - 35 = 4xB - 7B$$ $$ (24x^2 - 4x^2) + (-58x + 27x) + (28 - 35) = 4xB - 7B$$ $$20x^2 - 31x - 7 = 4xB - 7B$$ 9. **Factor out $$B$$ on the right side:** $$20x^2 - 31x - 7 = B(4x - 7)$$ 10. **Solve for $$B$$:** $$B = \frac{20x^2 - 31x - 7}{4x - 7}$$ **Final answer:** $$\boxed{B = \frac{20x^2 - 31x - 7}{4x - 7}}$$