Let's factor the trinomial $4x^2 + 11x - 3$ by following these steps:

Step 1: Identify the coefficients

• $a = 4$ (coefficient of $x^2$)
• $b = 11$ (coefficient of $x$)
• $c = -3$ (constant term)

Step 2: Find two numbers that multiply to $a \times c = 4 \times (-3) = -12$ and add to $b = 11$

We need to find two numbers that multiply to $-12$ and add to $11$. The numbers $12$ and $-1$ work because:

• $12 \times (-1) = -12$
• $12 + (-1) = 11$

Step 3: Rewrite the middle term using these two numbers

Rewrite $11x$ as $12x - x$: $4x^2 + 12x - x - 3$

Step 4: Factor by grouping

Group the terms into two pairs and factor out the greatest common factor (GCF) from each group: $(4x^2 + 12x) - (x + 3)$ Factor out the GCF from each group: $4x(x + 3) - 1(x + 3)$

Step 5: Factor out the common binomial factor

Now, factor out the common binomial factor $x + 3$: $(4x - 1)(x + 3)$

Final Answer

The factored form of the trinomial $4x^2 + 11x - 3$ is: $(4x - 1)(x + 3)$

Would you like more details or have any questions?

Here are some related questions:

1. How would you approach factoring a trinomial with a negative constant term?
2. Can you explain why the numbers 12 and -1 were chosen?
3. How would you check if the factorization is correct?
4. What if the middle term could not be split into factors that satisfy the conditions?
5. How does the sign of the constant term affect the factoring process?

Tip: When factoring, be mindful of the signs in the trinomial, as they significantly impact the factors you choose.