Final answer:
To factor the polynomial 6x²+29x+20, we look for two numbers that multiply to 120 (ac) and sum to 29 (b). The numbers are 5 and 24, leading us to the factored form: (6x+5)(x+4).
Step-by-step explanation:
To factor the polynomial 6x²+29x+20, we need to find two numbers that multiply to ac (where a is the coefficient of the x2 term and c is the constant term) and that also add up to b (the coefficient of the x term).
In this case, a is 6 and c is 20, so ac equals 120. We need two numbers that multiply to 120 and add up to 29, the coefficient of the middle term. Those two numbers are 5 and 24, since 5 * 24 = 120 and 5 + 24 = 29.
The polynomial can be rewritten splitting the middle term using the numbers we found:
6x²+24x+5x+20
Then we factor by grouping:
(6x²+24x) + (5x+20)
6x(x+4) + 5(x+4)
Finally, we factor out the common factor, which is (x+4), giving us the fully factored polynomial:
(6x+5)(x+4)