1. **Problem:** Simplify and expand the expressions given. 2. For the first expression: $$(4x^2 - 16x + 7)(4x^2 - 16x + 15) + 16$$ Use the distributive property (FOIL): $$(4x^2)(4x^2) + (4x^2)(-16x) + (4x^2)(15) + (-16x)(4x^2) + (-16x)(-16x) + (-16x)(15) + 7(4x^2) + 7(-16x) + 7(15) + 16$$ Calculate each: $$16x^4 - 64x^3 + 60x^2 - 64x^3 + 256x^2 - 240x + 28x^2 - 112x + 105 + 16$$ Combine like terms: $$16x^4 - 128x^3 + (60 + 256 + 28)x^2 + (-240 - 112)x + (105 + 16)$$ $$16x^4 - 128x^3 + 344x^2 - 352x + 121$$ 3. For the second expression: $$(9x^2 + 9x - 4)(9x^2 + 9x - 10) - 72$$ Multiply terms: $$(9x^2)(9x^2) + (9x^2)(9x) + (9x^2)(-10) + (9x)(9x^2) + (9x)(9x) + (9x)(-10) + (-4)(9x^2) + (-4)(9x) + (-4)(-10) - 72$$ Calculate each: $$81x^4 + 81x^3 - 90x^2 + 81x^3 + 81x^2 - 90x - 36x^2 - 36x + 40 - 72$$ Combine like terms: $$81x^4 + (81x^3 + 81x^3) + (-90x^2 + 81x^2 - 36x^2) + (-90x - 36x) + (40 - 72)$$ $$81x^4 + 162x^3 - 45x^2 - 126x - 32$$ 4. For the third expression: $$(x + 2)(x + 4)(x + 6)(x + 8) - 9$$ Pair for easier expansion: $$[(x + 2)(x + 8)] imes [(x + 4)(x + 6)] - 9$$ Expand each pair: $$ (x^2 + 10x + 16)(x^2 + 10x + 24) - 9$$ Multiply: $$ x^2 imes x^2 + x^2 imes 10x + x^2 imes 24 + 10x imes x^2 + 10x imes 10x + 10x imes 24 + 16 imes x^2 + 16 imes 10x + 16 imes 24 - 9$$ Calculate each: $$ x^4 + 10x^3 + 24x^2 + 10x^3 + 100x^2 + 240x + 16x^2 + 160x + 384 - 9$$ Combine terms: $$ x^4 + (10x^3 + 10x^3) + (24x^2 + 100x^2 + 16x^2) + (240x + 160x) + (384 - 9)$$ $$ x^4 + 20x^3 + 140x^2 + 400x + 375$$ 5. For the fourth expression: $$x(x + 1)(x + 2)(x + 3) + 1$$ Expand stepwise: $$x[(x + 1)(x + 2)(x + 3)] + 1$$ First, $$(x + 1)(x + 2) = x^2 + 3x + 2$$ Multiply with $(x + 3)$: $$(x^2 + 3x + 2)(x + 3) = x^3 + 3x^2 + 3x^2 + 9x + 2x + 6 = x^3 + 6x^2 + 11x + 6$$ Now multiply by $x$: $$x(x^3 + 6x^2 + 11x + 6) = x^4 + 6x^3 + 11x^2 + 6x$$ Add 1: $$x^4 + 6x^3 + 11x^2 + 6x + 1$$ 6. For the fifth expression: $$(x + 1)(x + 2)(x + 3)(x + 6) - 3x^2$$ Group: $$[(x + 1)(x + 6)] imes [(x + 2)(x + 3)] - 3x^2$$ Calculate each pair: $$(x^2 + 7x + 6)(x^2 + 5x + 6) - 3x^2$$ Multiply: $$(x^2)(x^2) + (x^2)(5x) + (x^2)(6) + 7x(x^2) + 7x(5x) + 7x(6) + 6(x^2) + 6(5x) + 6(6) - 3x^2$$ Calculate: $$x^4 + 5x^3 + 6x^2 + 7x^3 + 35x^2 + 42x + 6x^2 + 30x + 36 - 3x^2$$ Combine terms: $$x^4 + (5x^3 + 7x^3) + (6x^2 + 35x^2 + 6x^2 - 3x^2) + (42x + 30x) + 36$$ $$x^4 + 12x^3 + 44x^2 + 72x + 36$$ 7. For the sixth expression: $$64x^3 - 144x^2 y + 108xy^2 - 27y^3$$ Recognize it as a perfect cube expansion for the difference of cubes: $$(4x)^3 - 3(4x)^2(3y) + 3(4x)(3y)^2 - (3y)^3$$ Which equals: $$(4x - 3y)^3$$ **Final answers:** 1. $$16x^4 - 128x^3 + 344x^2 - 352x + 121$$ 2. $$81x^4 + 162x^3 - 45x^2 - 126x - 32$$ 3. $$x^4 + 20x^3 + 140x^2 + 400x + 375$$ 4. $$x^4 + 6x^3 + 11x^2 + 6x + 1$$ 5. $$x^4 + 12x^3 + 44x^2 + 72x + 36$$ 6. $$(4x - 3y)^3$$