How do you factor the expression 3x² + 10x + 8?

Factoring the Expression 3x² + 10x + 8

To factor the quadratic expression 3x² + 10x + 8, we can make use of the factoring technique that involves finding two numbers that multiply to the product of the leading coefficient (3) and the constant term (8), and also sum to the middle coefficient (10).

1. Identify the coefficients:
   The expression is in the form of ax² + bx + c, where:
   • a = 3
   • b = 10
   • c = 8
2. Calculate the product of a and c:
   Multiply the leading coefficient and the constant term:
   3 * 8 = 24
3. Find two numbers that fit the criteria:
   We need two numbers that multiply to 24 and add up to 10. The numbers that work are 4 and 6, because:
   4 * 6 = 24 and 4 + 6 = 10
4. Rewrite the middle term:
   Now, replace the middle term (10x) with 4x and 6x:
   3x² + 4x + 6x + 8
5. Group the terms:
   Now, group the terms in pairs:
   (3x² + 4x) + (6x + 8)
6. Factor by grouping:
   Factor out the common factors in each group:
   x(3x + 4) + 2(3x + 4)
7. Factor out the common binomial:
   Now, we can factor out the common binomial factor:
   
   (3x + 4)(x + 2)