Certainly! Let's expand the expression [tex]\(8(3a + 2)(5a - 3)\)[/tex] step by step.
### Step 1: Distribute the 8
First, keep the 8 outside for now and focus on expanding the binomials inside the parentheses.
[tex]\[ 8(3a + 2)(5a - 3) \][/tex]
### Step 2: Expand the binomials using the distributive property (FOIL method)
To expand [tex]\((3a + 2)(5a - 3)\)[/tex], we use the FOIL method which stands for:
- First terms
- Outer terms
- Inner terms
- Last terms
Let's identify each component:
#### First terms:
[tex]\[ (3a) \times (5a) = 15a^2 \][/tex]
#### Outer terms:
[tex]\[ (3a) \times (-3) = -9a \][/tex]
#### Inner terms:
[tex]\[ (2) \times (5a) = 10a \][/tex]
#### Last terms:
[tex]\[ (2) \times (-3) = -6 \][/tex]
Now, combine the like terms (middle terms: outer and inner):
[tex]\[ -9a + 10a = 1a \][/tex]
So, the expanded form of [tex]\((3a + 2)(5a - 3)\)[/tex] is:
[tex]\[ 15a^2 + 1a - 6 \][/tex]
### Step 3: Distribute the 8 to each term inside the parentheses
Now, multiply each term in the expanded polynomial by 8:
[tex]\[ 8 \times 15a^2 = 120a^2 \][/tex]
[tex]\[ 8 \times 1a = 8a \][/tex]
[tex]\[ 8 \times -6 = -48 \][/tex]
### Step 4: Combine all the terms
Putting it all together, we get the expanded polynomial:
[tex]\[ 120a^2 + 8a - 48 \][/tex]
So, the fully expanded form of [tex]\(8(3a + 2)(5a - 3)\)[/tex] is:
[tex]\[ 120a^2 + 8a - 48 \][/tex]
