Let's factorize the quadratic expression $4x^2 + x - 3$ into two linear factors.

Given: $4x^2 + x - 3$

To factorize a quadratic expression of the form $ax^2 + bx + c$, we look for two numbers that multiply to $ac$ (the product of the coefficient of $x^2$ and the constant term) and add to $b$ (the coefficient of $x$).

Here:

• $a = 4$
• $b = 1$
• $c = -3$

First, calculate $ac = 4 \times (-3) = -12$.

We need to find two numbers that multiply to $-12$ and add to $1$. These numbers are $4$ and $-3$ because: $4 \times (-3) = -12$ $4 + (-3) = 1$

Now, split the middle term $x$ using these two numbers: $4x^2 + 4x - 3x - 3$

Group the terms: $(4x^2 + 4x) - (3x + 3)$

Factor out the common factors from each group: $4x(x + 1) - 3(x + 1)$

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

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

Would you like more details on this, or do you have any questions? Here are some related questions that might interest you:

1. How do you identify the correct pair of factors when factorizing quadratics?
2. What are the methods for factorizing quadratic expressions?
3. Can all quadratic expressions be factorized into linear factors?
4. How do you solve a quadratic equation using factorization?
5. What is the quadratic formula, and how does it relate to factorization?

Tip: When factorizing quadratics, always double-check by expanding your factors to ensure they match the original expression.