1. \( x^{2}+2 x=3 \)

To solve the equation \( x^{2}+2x=3 \), start by rearranging it to the standard quadratic form: \( x^{2}+2x-3=0 \). Next, factor the quadratic expression. You need two numbers that multiply to -3 and add to 2; those numbers are 3 and -1. So, we can factor it as \( (x+3)(x-1)=0 \). Setting each factor to zero gives you the solutions: \( x+3=0 \) leads to \( x=-3 \), and \( x-1=0 \) leads to \( x=1 \). In this scenario, you can easily validate your solutions by substituting them back into the original equation. For \( x = -3 \): \( (-3)^{2} + 2(-3) = 9 - 6 = 3 \) is true. For \( x = 1 \): \( (1)^{2} + 2(1) = 1 + 2 = 3 \) also holds true. Both values work!