We need to factor the quadratic expression $7x^2 + 3x - 4$.

To factor a trinomial like this, we look for two numbers that multiply to give the product of the coefficient of $x^2$ and the constant term (i.e., $7 \times -4 = -28$) and that add up to the middle coefficient (i.e., 3). Let's proceed step by step:

Step 1: Find two numbers whose product is $-28$ and sum is $3$.

The two numbers that fit this are $7$ and $-4$, because:

• $7 \times (-4) = -28$
• $7 + (-4) = 3$

Step 2: Split the middle term using these two numbers.

We rewrite the equation as: $7x^2 + 7x - 4x - 4$

Step 3: Factor by grouping.

Now, we group the terms: $(7x^2 + 7x) - (4x + 4)$ Factor out the common terms in each group: $7x(x + 1) - 4(x + 1)$

Step 4: Factor out the common binomial factor.

Now, factor out $(x + 1)$: $(7x - 4)(x + 1)$

So the factored form of $7x^2 + 3x - 4$ is: $(7x - 4)(x + 1)$

The missing values in the equation are $7$ and $-4$ for the first bracket and $1$ for the second.

Would you like a more detailed explanation, or do you have any questions?

Here are 5 related questions to expand on this:

1. How do you factor a trinomial with a negative middle term?
2. Can every quadratic equation be factored?
3. How would you solve this quadratic equation after factoring?
4. What are the benefits of factoring in solving quadratic equations?
5. How can you check if the factorization is correct?

Tip: After factoring, you can always multiply the factors back out to check your work!