Factorize 3x^2 + 14x + 15

Sure, here are the steps to factorize the quadratic equation 3x^2 + 14x + 15:

Step 1: Identify the coefficients of the quadratic equation. Here, a = 3, b = 14, and c = 15.

Step 2: Find two numbers that multiply to give ac (in this case, 315 = 45) and add to give b (in this case, 14). The numbers that satisfy these conditions are 5 and 9, because 59 = 45 and 5+9 = 14.

Step 3: Rewrite the middle term of the quadratic equation as the sum of the terms 5x and 9x. This gives us 3x^2 + 5x + 9x + 15.

Step 4: Factor by grouping. The first two terms can be grouped together, and the last two terms can be grouped together. This gives us x(3x + 5) + 3(3x + 5).

Step 5: Notice that (3x + 5) is a common factor. Factor out this term to get the final factorized form of the quadratic equation: (3x + 5)(x + 3).

So, 3x^2 + 14x + 15 = (3x + 5)(x + 3).