What is the factored form of the quadratic expression 3y^2 + 7y + 4?

To determine the factored form of the quadratic expression (3y^2 + 7y + 4), let's first identify the product-sum relationship for the quadratic formula. We are looking for two numbers that multiply to the product of the leading coefficient (3) and the constant term (4), which is (3 \cdot 4 = 12), while also adding up to the middle coefficient (7).

The two numbers that satisfy this are 3 and 4, since:

• (3 \cdot 4 = 12)

• (3 + 4 = 7)

Using these two numbers, we can rewrite the middle term:

[

3y^2 + 3y + 4y + 4

]

Next, we group the terms:

[

(3y^2 + 3y) + (4y + 4)

]

This simplifies to:

[

3y(y + 1) + 4(y + 1)

]

Now we can factor out the common factor ((y + 1)):

[

(y + 1)(3y + 4)

]

This shows us that one valid factored form