`7a^3+56b^3`

Explanation

To factor the expression \(7a^3 + 56b^3\), we can follow these steps:

### Step 1: Identify the common factor
First, we observe that both terms \(7a^3\) and \(56b^3\) have a common factor. The greatest common factor (GCF) is \(7\).

### Step 2: Factor out the GCF
We factor out \(7\) from the expression:
\[ 7a^3 + 56b^3 = 7(a^3 + 8b^3) \]

### Step 3: Recognize the sum of cubes
Next, we notice that \(a^3 + 8b^3\) can be rewritten as \(a^3 + (2b)^3\). This fits the form of a sum of cubes, which is \(p^3 + q^3\) where \(p = a\) and \(q = 2b\).

### Step 4: Apply the sum of cubes formula
The formula for factoring a sum of cubes is:
\[ p^3 + q^3 = (p + q)(p^2 - pq + q^2) \]
Substituting \(p = a\) and \(q = 2b\):
\[ a^3 + (2b)^3 = (a + 2b)(a^2 - a(2b) + (2b)^2) \]

### Step 5: Simplify the expression
Now we simplify the second part:
\[ a^2 - a(2b) + (2b)^2 = a^2 - 2ab + 4b^2 \]
Putting it all together, we have:
\[ a^3 + 8b^3 = (a + 2b)(a^2 - 2ab + 4b^2) \]

### Step 6: Combine with the GCF
Finally, we combine this with the GCF we factored out earlier:
\[ 7(a^3 + 8b^3) = 7[(a + 2b)(a^2 - 2ab + 4b^2)] \]

### Final Answer
Thus, the fully factored form of the expression \(7a^3 + 56b^3\) is:
\[ \boxed{7(a + 2b)(a^2 - 2ab + 4b^2)} \]

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