a.	 Factorize:  (5x + y)2 - 11(5x + y) + 18 = 0.

We are given the quadratic expression:

\[
(5x + y)^2 - 11(5x + y) + 18 = 0
\]

Let us perform a substitution. Define \( z = 5x + y \). The equation becomes:

\[
z^2 - 11z + 18 = 0
\]

This is now a standard quadratic equation. To factorize it, we look for two numbers whose product is \( 18 \) and whose sum is \( -11 \). These numbers are \( -9 \) and \( -2 \), since:

\[
(-9) \times (-2) = 18 \quad \text{and} \quad (-9) + (-2) = -11
\]

So, we can factorize the quadratic as:

\[
z^2 - 11z + 18 = (z - 9)(z - 2)
\]

Substitute back \( z = 5x + y \):

\[
(5x + y - 9)(5x + y - 2) = 0
\]

Thus, the factorization of the original equation is:

\[
(5x + y - 9)(5x + y - 2) = 0
\]

### Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How do you solve a quadratic equation by factoring?
2. Can you factorize \( x^2 - 6x + 9 \)?
3. What are the steps to solve quadratic equations using the quadratic formula?
4. How would you expand \( (5x + y - 9)(5x + y - 2) \)?
5. How can you complete the square for \( z^2 - 11z + 18 = 0 \)?

**Tip**: Always check your factorization by expanding the factors back to ensure correctness.

(5x + y)^2 - 11(5x + y) + 18 = 0. Factorize.

Explore how to factorize the quadratic equation (5x + y)^2 - 11(5x + y) + 18 using substitution and factoring techniques. This solution breaks down algebraic concepts suitable for students in Grades 9-11, including the quadratic equation formula and the zero product property.

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