In the equation y = (x^2 + 2x + 1)/(x^2 + 4x + 3), what is the vertical asymptote?

To determine the vertical asymptote of the function given by the equation y = (x^2 + 2x + 1)/(x^2 + 4x + 3), it is essential to analyze the denominator of the fraction. Vertical asymptotes occur at values of x that make the denominator equal to zero, provided that these values do not also make the numerator zero (which would indicate a removable discontinuity instead).

First, let's factor the denominator, which is x^2 + 4x + 3. Factoring this expression, we can write it as (x + 1)(x + 3). This shows that the denominator equals zero when x + 1 = 0 or x + 3 = 0, which gives us the solutions x = -1 and x = -3.

The denominator will not equal zero, and thus there will be a vertical asymptote, at x = -3, where the function approaches infinity or negative infinity. The numerator, which is x^2 + 2x + 1, factors to (x + 1)(x + 1), and doesn’t equal zero at x = -3, affirming that this is indeed a vertical asym