https://file.aiquickdraw.com/mathbot/file/1726532393907jzla5pi9.jpg

Let's go over the two parts of the problem in the image:

### Part (a): Factoring the given quadratic expressions
You need to fill in the missing components of each factored form.

1. For the expression \(2x^2 + 11x + 5\), it needs to be factored. The general form of factoring looks like this:
   \[
   2x^2 + 11x + 5 = (2x + \_)(x + \_)
   \]
   To factor, look for two numbers that multiply to \(2 \times 5 = 10\) and add up to \(11\). The factors are \(10\) and \(1\). So, the factored form is:
   \[
   (2x + 1)(x + 5)
   \]

2. For the expression \(5x^2 - 18x + 9\), we use a similar process. The general form of the factorization is:
   \[
   5x^2 - 18x + 9 = (5x - \_)(x - \_)
   \]
   Here, we look for two numbers that multiply to \(5 \times 9 = 45\) and add up to \(-18\). The numbers are \(-15\) and \(-3\). So, the factored form is:
   \[
   (5x - 3)(x - 3)
   \]

### Part (b): Identifying the factors of \(2x^2 - 18x + 40\)

This quadratic expression can be factored. We start by factoring out the greatest common factor (GCF), which is 2:
\[
2x^2 - 18x + 40 = 2(x^2 - 9x + 20)
\]
Now, factor the quadratic \(x^2 - 9x + 20\). We need two numbers that multiply to \(20\) and add to \(-9\), which are \(-5\) and \(-4\). So the complete factorization is:
\[
2(x - 5)(x - 4)
\]
The factors of the expression are \(2\), \(x - 5\), and \(x - 4\).

### Answers for Part (b):
- \(C: (x - 5)\)
- \(J: (x - 4)\)
- \(G: 2\)

### Would you like any more details on any of the steps?

#### 5 Related Questions:
1. How do you find the greatest common factor (GCF) in a quadratic expression?
2. What is the process for factoring quadratics when the coefficient of \(x^2\) is not 1?
3. Can all quadratic equations be factored by simple methods like these?
4. What if you cannot find factors for the quadratic expression — what should you do next?
5. How does completing the square differ from factoring for solving quadratics?

#### Tip:
Always check your factored expression by expanding (foiling) it back out to confirm it equals the original equation!

Fill in the missing components in each factored form and identify the factors of the quadratic expression.

Learn how to factor the quadratic expressions 2x^2 + 11x + 5 and 5x^2 - 18x + 9. This problem explains how to fill in the missing components and covers important concepts like factoring using the greatest common factor (GCF) and the factorization theorem. Suitable for Grades 9-10 students.

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