Cubic and Quartics, solving methods?

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Cubic and quartic equations can be solved using various techniques beyond lengthy formulas. Numerical methods, such as the Newton-Raphson method, are recommended for finding roots efficiently. While it's useful to have the formulas for reference, focusing on practical solving techniques can streamline the process. The discussion highlights the importance of effective problem-solving strategies in mathematics. Overall, exploring alternative methods can enhance understanding and application of cubic and quartic equations.
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Instead of long ass "formulas" for Cubic and Quartics equations, can someone give some solving techniques? Perhaps in the end attach the formula just in case your hand gets itchy?
 
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Edit: This post is a reply to an impolite comment by Prudens that has since been deleted.
 
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Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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