The given equation is:

$(a + b)(a + b + c) = a^2 + 2ab + b^2 + ac + bc$

We will verify if the left-hand side expands correctly into the right-hand side:

Step 1: Expand the left-hand side

$(a + b)(a + b + c) = (a + b)a + (a + b)b + (a + b)c$

Expanding each term:

• $(a + b)a = a^2 + ab$
• $(a + b)b = ab + b^2$
• $(a + b)c = ac + bc$

Adding these terms together: $a^2 + ab + ab + b^2 + ac + bc = a^2 + 2ab + b^2 + ac + bc$

Step 2: Compare with the right-hand side

The expanded left-hand side matches the right-hand side exactly: $a^2 + 2ab + b^2 + ac + bc$

Conclusion:

The equation is True.

Would you like further explanation? Here are some related questions to deepen your understanding:

1. How do you identify the distributive property when expanding brackets?
2. What are common errors to avoid during expansion of algebraic expressions?
3. Can you generalize this method to larger polynomials?
4. How can factoring be used to verify an expanded equation?
5. What is the significance of combining like terms in such problems?

Tip: When expanding algebraic expressions, carefully distribute each term in the first set of parentheses to every term in the second.