1. **State the problem:** Simplify the expression $$(x+1)(x+3)(x+5)(x+7) + 15$$. 2. **Use the formula and rules:** We will first multiply the factors in pairs to simplify the expression step-by-step. 3. **Multiply the first two binomials:** $$ (x+1)(x+3) = x^2 + 3x + x + 3 = x^2 + 4x + 3 $$ 4. **Multiply the next two binomials:** $$ (x+5)(x+7) = x^2 + 7x + 5x + 35 = x^2 + 12x + 35 $$ 5. **Multiply the two quadratic expressions:** $$ (x^2 + 4x + 3)(x^2 + 12x + 35) $$ Use distributive property: $$ = x^2(x^2 + 12x + 35) + 4x(x^2 + 12x + 35) + 3(x^2 + 12x + 35) $$ $$ = x^4 + 12x^3 + 35x^2 + 4x^3 + 48x^2 + 140x + 3x^2 + 36x + 105 $$ 6. **Combine like terms:** $$ x^4 + (12x^3 + 4x^3) + (35x^2 + 48x^2 + 3x^2) + (140x + 36x) + 105 $$ $$ = x^4 + 16x^3 + 86x^2 + 176x + 105 $$ 7. **Add the constant 15:** $$ x^4 + 16x^3 + 86x^2 + 176x + 105 + 15 = x^4 + 16x^3 + 86x^2 + 176x + 120 $$ **Final answer:** $$ x^4 + 16x^3 + 86x^2 + 176x + 120 $$