Multiply :
`2y - 4y^(3) + 6y^(5)` by `y^(2) + y - 3`

Answer

To solve the problem of multiplying the algebraic expressions \(2y - 4y^3 + 6y^5\) by \(y^2 + y - 3\), we will follow these steps:

### Step 1: Distribute Each Term
We will distribute each term from the first polynomial \(2y - 4y^3 + 6y^5\) to each term in the second polynomial \(y^2 + y - 3\).

### Step 2: Multiply \(2y\) by Each Term
- \(2y \cdot y^2 = 2y^3\)
- \(2y \cdot y = 2y^2\)
- \(2y \cdot (-3) = -6y\)

### Step 3: Multiply \(-4y^3\) by Each Term
- \(-4y^3 \cdot y^2 = -4y^5\)
- \(-4y^3 \cdot y = -4y^4\)
- \(-4y^3 \cdot (-3) = 12y^3\)

### Step 4: Multiply \(6y^5\) by Each Term
- \(6y^5 \cdot y^2 = 6y^7\)
- \(6y^5 \cdot y = 6y^6\)
- \(6y^5 \cdot (-3) = -18y^5\)

### Step 5: Combine All the Results
Now we combine all the results from the previous steps:
- From \(2y\): \(2y^3 + 2y^2 - 6y\)
- From \(-4y^3\): \(-4y^5 - 4y^4 + 12y^3\)
- From \(6y^5\): \(6y^7 + 6y^6 - 18y^5\)

### Step 6: Write All Terms Together
Combining all terms we have:
\[ 6y^7 + 6y^6 + (2y^3 + 12y^3) + (-4y^5 - 18y^5) - 4y^4 + 2y^2 - 6y \]

### Step 7: Combine Like Terms
Now we combine like terms:
- \(6y^7\)
- \(6y^6\)
- \( (2y^3 + 12y^3) = 14y^3\)
- \( (-4y^5 - 18y^5) = -22y^5\)
- \(-4y^4\)
- \(2y^2\)
- \(-6y\)

### Final Result
Arranging in decreasing order of powers:
\[ 6y^7 + 6y^6 - 22y^5 - 4y^4 + 14y^3 + 2y^2 - 6y \]