Solve the Quadratic Equation: x² + x - 2 = 0 Step by Step

Our goal is to factor the expression on the left side of the given equation:

$x^2+x-2=0$

Note that the coefficient of the quadratic term in the expression on the left side is 1, therefore, we can (try to) factor the expression by using quick trinomial factoring:

Let's look for a pair of numbers whose product equals the free term in the expression, and whose sum equals the coefficient of the first-degree term, meaning two numbers $m,\hspace{2pt}n$ that satisfy the given values:

$m\cdot n=-2\\ m+n=1\\$From the first requirement mentioned, that is - from the multiplication, we notice that the product of the numbers we're looking for needs to be negative. Therefore we can conclude that the two numbers have different signs, according to the multiplication rules. Note that the possible factors of 2 are 2 and 1, fulfilling the second requirement mentioned. Furthermore the fact that the signs of the numbers are different from each other leads us to the conclusion that the only possibility for the two numbers we're looking for is:

$\begin{cases} m=-1\\ n=2 \end{cases}$

Therefore we can factor the expression on the left side of the equation to:

$x^2+x-2=0 \\ \downarrow\\ (x-1)(x+2)=0$

The correct answer is answer A.