1. The problem is to simplify the expression $$(x+h) [4(x+h)^2 + (x+h) -1] -4x^3 - x^2 + x$$ after expanding and distributing. 2. First, expand the terms inside the brackets: $$4(x+h)^2 + (x+h) -1$$ Recall that $$(x+h)^2 = x^2 + 2xh + h^2$$ So, $$4(x+h)^2 = 4(x^2 + 2xh + h^2) = 4x^2 + 8xh + 4h^2$$ 3. Now add the remaining terms inside the bracket: $$4x^2 + 8xh + 4h^2 + x + h - 1$$ 4. Multiply this entire expression by $$(x+h)$$: $$(x+h)(4x^2 + 8xh + 4h^2 + x + h - 1)$$ Distribute each term: $$x(4x^2) + x(8xh) + x(4h^2) + x(x) + x(h) - x(1) + h(4x^2) + h(8xh) + h(4h^2) + h(x) + h(h) - h(1)$$ Simplify each: $$4x^3 + 8x^2h + 4xh^2 + x^2 + xh - x + 4x^2h + 8xh^2 + 4h^3 + xh + h^2 - h$$ 5. Combine like terms: - $4x^3$ - $8x^2h + 4x^2h = 12x^2h$ - $4xh^2 + 8xh^2 = 12xh^2$ - $x^2$ - $xh + xh = 2xh$ - $4h^3$ - $h^2$ - $-x$ - $-h$ So the expanded form is: $$4x^3 + 12x^2h + 12xh^2 + x^2 + 2xh + 4h^3 + h^2 - x - h$$ 6. Now subtract the remaining terms from the original expression: $$-4x^3 - x^2 + x$$ Add these to the expanded expression: $$4x^3 + 12x^2h + 12xh^2 + x^2 + 2xh + 4h^3 + h^2 - x - h - 4x^3 - x^2 + x$$ 7. Combine like terms again: - $4x^3 - 4x^3 = 0$ - $x^2 - x^2 = 0$ - $-x + x = 0$ Remaining terms: $$12x^2h + 12xh^2 + 2xh + 4h^3 + h^2 - h$$ 8. Final simplified expression: $$12x^2h + 12xh^2 + 2xh + 4h^3 + h^2 - h$$