1. **State the problem:** Simplify the expression $$(3-x)^2 (4+x)^2$$. 2. **Rewrite the expression:** Notice this is a product of two squared binomials: $$((3-x)(4+x))^2$$ 3. **Expand the inner product:** Use FOIL: $$(3-x)(4+x) = 3 \times 4 + 3 \times x - x \times 4 - x \times x = 12 + 3x - 4x - x^2$$ 4. **Simplify inside:** $$12 + 3x - 4x - x^2 = 12 - x - x^2$$ 5. **Rearrange into standard polynomial form:** $$-x^2 - x + 12$$ 6. **Square the result:** $$(-(x^2 + x - 12))^2 = (x^2 + x - 12)^2$$ 7. **Expand the square:** Use the formula $$(a+b+c)^2 = a^2 + 2ab + 2ac + b^2 + 2bc + c^2$$ where $a = x^2$, $b = x$, $c = -12$: $$ (x^2)^2 + 2 \times x^2 \times x + 2 \times x^2 \times (-12) + x^2 + 2 \times x \times (-12) + (-12)^2$$ Compute each term: $$ x^4 + 2x^3 - 24x^2 + x^2 -24x + 144$$ 8. **Combine like terms:** $$x^4 + 2x^3 - 23x^2 - 24x + 144$$ **Final answer:** $$ (3 - x)^2 (4 + x)^2 = x^4 + 2x^3 - 23x^2 - 24x + 144 $$