Factor completely \( x^{2}+8 x+12 \)

To factor the quadratic expression \( x^{2}+8x+12 \), we first look for two numbers that multiply to 12 (the constant term) and add up to 8 (the coefficient of \( x \)). Those numbers are 2 and 6! So, we can write: \( x^{2}+8x+12 = (x+2)(x+6) \). This shows that the expression factors neatly into the product of two binomials. Perfectly simple, isn't it? Now let’s verify by multiplying the factors back together: \((x+2)(x+6) = x^2 + 6x + 2x + 12 = x^2 + 8x + 12\). The original quadratic is indeed the result!