Factor completely with solution   1) 2y²+2    2) m³+343             3) 3m³+24

Let's go step by step and factor each expression completely:

### 1) \( 2y^2 + 2 \)

This expression has a common factor of \( 2 \), so factor that out first:

\[
2y^2 + 2 = 2(y^2 + 1)
\]

Since \( y^2 + 1 \) cannot be factored further over the real numbers, the completely factored form is:

\[
\boxed{2(y^2 + 1)}
\]

---

### 2) \( m^3 + 343 \)

Recognize that 343 is a cube: \( 343 = 7^3 \). This expression is a sum of cubes:

\[
m^3 + 343 = m^3 + 7^3
\]

The sum of cubes formula is:

\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]

In this case, \( a = m \) and \( b = 7 \). Applying the formula:

\[
m^3 + 7^3 = (m + 7)(m^2 - 7m + 49)
\]

So the completely factored form is:

\[
\boxed{(m + 7)(m^2 - 7m + 49)}
\]

---

### 3) \( 3m^3 + 24 \)

First, factor out the greatest common factor (GCF), which is \( 3 \):

\[
3m^3 + 24 = 3(m^3 + 8)
\]

Next, recognize that \( 8 = 2^3 \), so \( m^3 + 8 \) is a sum of cubes:

\[
m^3 + 8 = m^3 + 2^3
\]

Again, use the sum of cubes formula:

\[
m^3 + 2^3 = (m + 2)(m^2 - 2m + 4)
\]

Substitute this back into the factored form:

\[
3(m^3 + 8) = 3(m + 2)(m^2 - 2m + 4)
\]

So the completely factored form is:

\[
\boxed{3(m + 2)(m^2 - 2m + 4)}
\]

---

### Do you want details or have any questions?

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**Related Questions:**

1. How do you factor the difference of cubes, like \( a^3 - b^3 \)?
2. What is the formula for factoring perfect square trinomials?
3. How do you recognize if an expression can be factored as a sum or difference of cubes?
4. What is the process for factoring polynomials with common factors first?
5. Can all quadratic expressions be factored over the real numbers?

**Tip:** Always look for a common factor first before attempting any more complex factoring techniques.

Factor completely with solution 1) 2y²+2 2) m³+343 3) 3m³+24

Learn how to factor expressions such as 2y²+2, m³+343, and 3m³+24 using the sum of cubes formula and greatest common factor (GCF). This solution breaks down the steps with detailed explanations, suitable for students in Grades 9-12.

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