Multiply :
`5p^(2) + 25pq + 4q^(2)` by `2p^(2) - 2pq + 3q^(2)`

Answer

To multiply the algebraic expressions \(5p^2 + 25pq + 4q^2\) and \(2p^2 - 2pq + 3q^2\), we will use the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step solution:

### Step 1: Write down the expressions
We have:
\[ (5p^2 + 25pq + 4q^2) \times (2p^2 - 2pq + 3q^2) \]

### Step 2: Distribute each term in the first expression to each term in the second expression
We will multiply each term in the first expression by each term in the second expression.

1. **Multiply \(5p^2\) by each term in the second expression:**
- \(5p^2 \times 2p^2 = 10p^4\)
- \(5p^2 \times (-2pq) = -10p^3q\)
- \(5p^2 \times 3q^2 = 15p^2q^2\)

2. **Multiply \(25pq\) by each term in the second expression:**
- \(25pq \times 2p^2 = 50p^3q\)
- \(25pq \times (-2pq) = -50p^2q^2\)
- \(25pq \times 3q^2 = 75pq^3\)

3. **Multiply \(4q^2\) by each term in the second expression:**
- \(4q^2 \times 2p^2 = 8p^2q^2\)
- \(4q^2 \times (-2pq) = -8pq^3\)
- \(4q^2 \times 3q^2 = 12q^4\)

### Step 3: Combine all the products
Now we will combine all the products we obtained from the multiplication:
\[ 10p^4 + (-10p^3q) + 15p^2q^2 + 50p^3q + (-50p^2q^2) + 75pq^3 + 8p^2q^2 + (-8pq^3) + 12q^4 \]

### Step 4: Group like terms
Now, we will group the like terms:
- \(p^4\) terms: \(10p^4\)
- \(p^3q\) terms: \(-10p^3q + 50p^3q = 40p^3q\)
- \(p^2q^2\) terms: \(15p^2q^2 - 50p^2q^2 + 8p^2q^2 = -27p^2q^2\)
- \(pq^3\) terms: \(75pq^3 - 8pq^3 = 67pq^3\)
- \(q^4\) term: \(12q^4\)

### Step 5: Write the final expression
Combining all these gives us:
\[ 10p^4 + 40p^3q - 27p^2q^2 + 67pq^3 + 12q^4 \]

### Final Answer:
Thus, the product of the two expressions is:
\[ \boxed{10p^4 + 40p^3q - 27p^2q^2 + 67pq^3 + 12q^4} \]