To factor the expression 25a² + 30a + 9 completely, we first look for two numbers that multiply to the product of the coefficient of a² (which is 25) and the constant term (which is 9). This total is 25 * 9 = 225. We also need these two numbers to add up to the coefficient of a (which is 30).
The two numbers that satisfy both conditions are 25 and 5 because:
- 25 * 5 = 225
- 25 + 5 = 30
Now, we can rewrite the middle term (30a) using these numbers:
25a² + 25a + 5a + 9
Next, we group the terms:
(25a² + 25a) + (5a + 9)
Now, we factor out the common factors from each group:
- From 25a² + 25a, we can factor out 25a, giving us 25a(a + 1).
- From 5a + 9, we factor out 1, giving us (5a + 9).
So, we have:
25a(a + 1) + 1(5a + 9)
Now we can put the expression in its factored form:
(25a + 9)(a + 1)
In conclusion, the fully factored form of the expression 25a² + 30a + 9 is:
(25a + 9)(a + 1)
Make sure to place these factors in the respective columns of your grid by following the corresponding sections for each factor.