To factor the quadratic expression 3x² + 8x + 5, we seek two binomials that multiply to give the original expression. The general form of factoring a quadratic equation is:
- ax² + bx + c = (px + q)(rx + s)
In this case, a = 3, b = 8, and c = 5.
Next, we need to find two numbers that multiply to a imes c = 3 imes 5 = 15 and add up to b = 8. The two numbers that satisfy these conditions are 3 and 5 since 3 imes 5 = 15 and 3 + 5 = 8.
Now, we rewrite the middle term (8x) using these two numbers:
3x² + 3x + 5x + 5
Next, we group the terms:
Group 1: 3x² + 3x and Group 2: 5x + 5
Now, we can factor each group:
- From Group 1: 3x(x + 1)
- From Group 2: 5(x + 1)
Now we have:
3x(x + 1) + 5(x + 1)
Notice that (x + 1) is common in both terms. We can factor that out:
(x + 1)(3x + 5)
Thus, the factorization of the expression 3x² + 8x + 5 is:
(x + 1)(3x + 5)
To confirm, you can distribute:
(x + 1)(3x + 5) = 3x² + 5x + 3x + 5 = 3x² + 8x + 5.
In conclusion, the factored form of the expression 3x² + 8x + 5 is (x + 1)(3x + 5).