https://file.aiquickdraw.com/mathbot/file/1726273212581qrjeu02t.jpg

The expression given in the image is:

\[
x^4 + 4x^3 - 7x^2 - 22x + 24
\]

We are tasked with factorizing this polynomial.

### Steps to factorize:

1. **Attempt Rational Root Test**:
   To factor, we first apply the Rational Root Theorem to check for possible rational roots. The possible rational roots are factors of the constant term (24) divided by factors of the leading coefficient (1). Hence, the possible rational roots are:
   
   \[
   \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24
   \]

2. **Test Possible Roots**:
   We substitute these values into the polynomial to see if any yield zero. Once we find a root, we can use polynomial division to factor it out.

3. **Continue Factorization**:
   After dividing the polynomial by the factor associated with the root, we continue factorizing the resulting polynomial until fully factored.

Let me proceed with testing for rational roots and the division steps.The polynomial \( x^4 + 4x^3 - 7x^2 - 22x + 24 \) has the following roots:

\[
x = 2, \quad x = 1, \quad x = -3, \quad x = -4
\]

Thus, the factorization of the polynomial is:

\[
(x - 2)(x - 1)(x + 3)(x + 4)
\]

This is the fully factorized form of the given polynomial.

Would you like more details or have any questions? Here are some related questions you might find helpful:

1. How do we perform synthetic division to confirm the factors?
2. What is the Rational Root Theorem, and how is it applied?
3. Can we factorize this polynomial using other methods, such as completing the square?
4. How do we factor polynomials with complex or irrational roots?
5. What are some real-life applications of polynomial factorization?

**Tip:** When factoring polynomials, always start by testing simple values using the Rational Root Theorem before moving to more complex techniques.

Learn how to factorize the polynomial x^4 + 4x^3 - 7x^2 - 22x + 24. This step-by-step guide covers important concepts like the Rational Root Theorem and polynomial division, providing a solution suitable for Grades 10-12.

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