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Understanding cross multiplication is essential in solving proportions, where two ratios are set equal to each other. It's a method used to eliminate the fractions from a proportion or to compare two fractions without finding a common denominator.

For example, consider the proportion \(\frac{a}{b} = \frac{c}{d}\). To solve for a variable, you cross multiply, resulting in \(ad = bc\). This technique gives you an equation without fractions, which is often easier to solve. It's particularly helpful in solving polynomial equations, as seen in the step-by-step solution where the numerator of one fraction is multiplied by the denominator of the other, and vice versa.

In the given exercise, cross multiplication transforms the complex fractional equation into a polynomial equation, paving the way to find the value of \(x\) by further algebraic manipulations.

Polynomial equations are mathematical expressions that involve a sum of powers in one or more variables. In the context of the exercise, the polynomial in question is a cubic equation, which is a particular type of polynomial equation where the largest exponent is 3.

The general form of a polynomial equation is \(a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0 = 0\), where \(a_n\) to \(a_0\) are constants. To solve these equations, one must combine like terms, rearrange the equation, and apply various factoring techniques if possible. When equations become complex, numerical methods or polynomial theorems may be necessary to find the roots.

When solving polynomial equations, it's often necessary to expand polynomials. Expansion involves multiplying out brackets to simplify the expression. This process uses the distributive property, where you multiply each term inside a set of parentheses by every term in another set.

For instance, the expression \( (x + 2)(x + 3) \) expands to \( x^2 + 3x + 2x + 6 \), which then simplifies to \( x^2 + 5x + 6 \). This is what occurs in our exercise example during the cross multiplication step, where each term of one polynomial is multiplied by every term of the other polynomial, leading to an expanded form that is a single polynomial equation to be solved.

A cubic equation is a type of polynomial equation that contains a variable raised to the third power as its highest degree. The standard form of a cubic equation is \(ax^3 + bx^2 + cx + d = 0\), where \(a\), \(b\), \(c\), and \(d\) are coefficients and \(a \eq 0\).

Solving cubic equations can be challenging. It may involve factoring by grouping, synthetic division, or the use of the cubic formula. In some cases, like the one in our exercise, factoring may not lead to a clear solution, and one might need to resort to numerical methods such as Newton's method, or software tools that can handle complex roots.