Completely factor the given trinomial.
 

8a2b−44ab+20b8a2b−44ab+20b




To factor the trinomial \( 8a^2b - 44ab + 20b \), follow these steps:

### Step 1: Find the greatest common factor (GCF)
The common factor among all the terms is \( 4b \).

Factor out the GCF:
\[
8a^2b - 44ab + 20b = 4b(2a^2 - 11a + 5)
\]

### Step 2: Factor the quadratic expression \( 2a^2 - 11a + 5 \)
We need to factor the quadratic expression \( 2a^2 - 11a + 5 \). Look for two numbers that multiply to \( 2 \times 5 = 10 \) and add up to \( -11 \). The numbers are \( -10 \) and \( -1 \).

Now, split the middle term using these numbers:
\[
2a^2 - 10a - a + 5
\]

### Step 3: Group the terms
Group the terms and factor:
\[
(2a^2 - 10a) - (a - 5)
\]
Factor each group:
\[
2a(a - 5) - 1(a - 5)
\]

### Step 4: Factor out the common binomial
Now, factor out \( (a - 5) \):
\[
(2a - 1)(a - 5)
\]

### Step 5: Write the complete factorization
The completely factored form of the trinomial is:
\[
4b(2a - 1)(a - 5)
\]

Would you like further details on any of the steps? Here are some related questions for deeper exploration:

1. What is the general method to factor trinomials of the form \( ax^2 + bx + c \)?
2. How do you find the greatest common factor (GCF) of terms?
3. Can trinomials always be factored, or are there cases when factoring isn't possible?
4. What is the importance of factoring in solving quadratic equations?
5. How can grouping terms be used in other factoring problems?

**Tip:** When factoring, always start by checking for a GCF before attempting more complex methods like grouping or using the quadratic formula.

Completely factor the given trinomial: 8a^2b - 44ab + 20b

This guide explains how to completely factor the trinomial 8a^2b - 44ab + 20b by using the GCF and factoring by grouping methods. Perfect for students in Grades 9-11, this step-by-step solution covers algebraic concepts such as the greatest common factor and quadratic equations.

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