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The problem in the image reads:

"If \( x^4 + px^2 + q \) has a factor of \( x^2 - 4x + 3 \), find \( p \) and \( q \)."

### Solution:
Given that \( x^4 + px^2 + q \) has a factor \( x^2 - 4x + 3 \), we can express \( x^4 + px^2 + q \) as a product of this factor and another quadratic polynomial:

\[
x^4 + px^2 + q = (x^2 - 4x + 3)(x^2 + ax + b)
\]

To find the values of \( p \) and \( q \), we expand the product on the right-hand side:

\[
(x^2 - 4x + 3)(x^2 + ax + b) = x^4 + ax^3 + bx^2 - 4x^3 - 4ax^2 - 4bx + 3x^2 + 3ax + 3b
\]

Combining like terms:

\[
x^4 + (a - 4)x^3 + (b - 4a + 3)x^2 + (3a - 4b)x + 3b
\]

Since there is no \( x^3 \) or \( x \) term in \( x^4 + px^2 + q \), we must have:

\[
a - 4 = 0 \quad \text{and} \quad 3a - 4b = 0
\]

Solving these, we first find \( a \):

\[
a = 4
\]

Next, substitute \( a = 4 \) into \( 3a - 4b = 0 \):

\[
3(4) - 4b = 0 \implies 12 = 4b \implies b = 3
\]

Now substitute \( a = 4 \) and \( b = 3 \) into the coefficient for \( x^2 \) to find \( p \):

\[
p = b - 4a + 3 = 3 - 4(4) + 3 = 3 - 16 + 3 = -10
\]

Finally, \( q \) is given by:

\[
q = 3b = 3(3) = 9
\]

### Final Answer:
\[
p = -10 \quad \text{and} \quad q = 9
\]

Would you like more details on any part of the solution, or do you have any questions?

Here are some related questions for further practice:

1. How would you factor \( x^4 + px^2 + q \) given \( p = -5 \) and \( q = 6 \)?
2. What is the general approach to finding factors of polynomials?
3. Can you derive a different pair \( p \) and \( q \) by altering the factor \( x^2 - 4x + 3 \)?
4. What is the role of the discriminant in factorizing quadratic equations?
5. How would the problem change if the polynomial had a different factor, say \( x^2 + x - 1 \)?

**Tip:** When given a polynomial with an unknown factor, expanding the product of polynomials is a systematic way to identify unknown coefficients.

Learn how to factorize the polynomial x^4 + px^2 + q using the given polynomial factor x^2 - 4x + 3. Detailed step-by-step solution provided, covering key concepts in polynomials and quadratic equations. Suitable for high school students.

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