A trinomial expression is any polynomial expression that has exactly three terms. In most cases, "solving" means factoring the expression out into its simplest components. Usually, your trinomial will either be a quadratic equation, or a higher-order equation that can be turned into a quadratic equation by factoring out variables common to all terms. Start by learning how to factor quadratics, then learn how to tackle other sorts of trinomials.
Factor out any factors common to all terms. The equation 4x^2 + 8x + 4 has 4 as a common factor, since every term can be divided by 4. Therefore, it can be factored as 4(x^2 + 2x +1). The equation x^3 +2x^2 + x has x as a common factor. It can be factored as x(x^2 +2x +1).
Look for any other common factors you may have missed. Sometimes, an equation has both a number and a variable that can be factored out. For example, 8x^3 +12x^2 + 16x has both 4 and x as a factor. Factored out, it becomes 4x(2x^2 + 3x + 4)
Determine what sort of trinomial equation you have left. If the highest power of the unfactored part is a squared variable like y^2 or 4a^2, you can factor it like a quadratic equation. If your highest power term is a cubed number or higher, you have a higher order equation. By this point, you will probably not have anything greater than a cubed variable to deal with.
Factor out the quadratic part of the equation. Many trinomial quadratics are simple sums of squares. Using an example from step one:
4x^2 + 8x + 4 = 4(x^2 + 2x + 1) = 4(x + 1)(x + 1) 4(x + 1)^2
If you are dealing with a higher-order equation, look for a pattern that allows you to solve it like a quadratic. For example, although 4x^4 + 12x^2 + 9 looks like a tough equation at first, the answer is actually very simple: 4x^4 + 12x^2 + 9 = (2x^2 + 3)^2
If you are dealing with a quadratic equation that you cannot factor, you can always apply the quadratic formula (see Resources).
Learn how to solve quadratic equations before trying to tackle harder trinomials. Quadratics will teach you the patterns you need to look for in more difficult equations.