(x+3)(x+1)(x+5)(x+7)+16=0

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To solve this equation, we can start by expanding the left side:

(x+3)(x+1)(x+5)(x+7) + 16 = 0

Expanding the left side:

(x^2 + 4x + 3)(x^2 + 12x + 35) + 16 = 0

Now, multiply the two quadratic expressions:

x^4 + 16x^3 + 71x^2 + 140x + 105 + 16 = 0

Combine like terms:

x^4 + 16x^3 + 71x^2 + 140x + 121 = 0

Now the equation is in the form of a quartic equation which can be difficult to solve. One way to solve it is by factoring or using numerical methods.