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Cross multiplication is a method often used to solve equations that involve fractions. It's a straightforward technique where you multiply diagonally across the equality of two fractions to eliminate the denominators, making the equation easier to work with.

Let's take the given exercise as an example. We have: \( \frac{x(x+5)}{4x+4} = \frac{9}{5} \). To apply cross multiplication, we multiply the numerator of the first fraction by the denominator of the second fraction and vice versa, so the equation becomes: \( 5x(x + 5) = 9(4x + 4) \). This step effectively removes the fractions and gives us a clearer pathway to finding the value of \( x \).

The distributive property is a critical algebraic property that allows us to multiply a single term across a sum or difference within parentheses. For example, in the equation we acquired after cross multiplication \( 5x(x + 5) = 9(4x + 4) \), we can distribute the \( 5 \) and \( 9 \) into the parentheses. The equation transforms into: \( 5x^2 + 25x = 36x + 36 \).

The distributive property simplifies the expression and helps set the stage for further manipulation, such as combining like terms or further simplification. In many instances, like this problem, the distributive property is a key step in solving quadratic equations.

In the case of our exercise, after simplification and rearrangement, we were left with a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), with \( a = 5 \), \( b = -11 \) and \( c = -36 \).

The quadratic formula, \( x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a} \), gives us a reliable method to find the values of \( x \) that satisfy the equation. By substituting our specific values into the formula, we can find the solutions to the equation: \( x = 4 \) and \( x = -1.8 \). The ± symbol in the formula indicates that two solutions exist, as it accounts for both the positive and negative square roots. This powerful tool is pivotal when factoring a quadratic isn't feasible or straightforward.