1. **State the problem:** We need to find the expression for the top base $B$ of a trapezoid given the area $A = 12x^2 - 29x + 14$, the bottom base $b = x - 5$, and the height $h = 4x - 7$. The area formula is: $$A = \frac{(B + b) \cdot h}{2}$$ 2. **Write the formula with given values:** $$12x^2 - 29x + 14 = \frac{(B + (x - 5)) \cdot (4x - 7)}{2}$$ 3. **Multiply both sides by 2 to clear the denominator:** $$2(12x^2 - 29x + 14) = (B + x - 5)(4x - 7)$$ which simplifies to $$24x^2 - 58x + 28 = (B + x - 5)(4x - 7)$$ 4. **Expand the right side:** $$(B + x - 5)(4x - 7) = B(4x - 7) + (x - 5)(4x - 7)$$ 5. **Expand $(x - 5)(4x - 7)$:** $$x \cdot 4x = 4x^2$$ $$x \cdot (-7) = -7x$$ $$-5 \cdot 4x = -20x$$ $$-5 \cdot (-7) = 35$$ Sum these: $$4x^2 - 7x - 20x + 35 = 4x^2 - 27x + 35$$ 6. **Rewrite the equation:** $$24x^2 - 58x + 28 = 4xB - 7B + 4x^2 - 27x + 35$$ 7. **Group terms and isolate terms with $B$:** $$24x^2 - 58x + 28 - 4x^2 + 27x - 35 = 4xB - 7B$$ Simplify left side: $$20x^2 - 31x - 7 = B(4x - 7)$$ 8. **Solve for $B$ by dividing both sides by $(4x - 7)$:** $$B = \frac{20x^2 - 31x - 7}{4x - 7}$$ **Final answer:** $$\boxed{B = \frac{20x^2 - 31x - 7}{4x - 7}}$$