Let's expand and simplify each expression step by step:

a) $(x - 3)(x + 7)$

We can apply the distributive property (also known as FOIL for binomials) to expand:

$(x - 3)(x + 7) = x(x + 7) - 3(x + 7)$ $= x^2 + 7x - 3x - 21$ $= x^2 + 4x - 21$

b) $(x + 2)^2$

To expand a binomial squared, we use the formula $(a + b)^2 = a^2 + 2ab + b^2$:

$(x + 2)^2 = x^2 + 2(2)(x) + 2^2$ $= x^2 + 4x + 4$

c) $(2x + 4)(x + 6)$

Again, we apply the distributive property:

$(2x + 4)(x + 6) = 2x(x + 6) + 4(x + 6)$ $= 2x^2 + 12x + 4x + 24$ $= 2x^2 + 16x + 24$

Final Answers:

• a) $x^2 + 4x - 21$
• b) $x^2 + 4x + 4$
• c) $2x^2 + 16x + 24$

Let me know if you'd like further clarification on any step!

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Relative Questions:

1. What is the distributive property in algebra?
2. Can you explain the FOIL method for binomials?
3. How do you expand a binomial square?
4. Why does the order of terms in a multiplication not matter?
5. What is the difference between factoring and expanding?

Tip: When expanding polynomials, organizing like terms before simplifying can help avoid mistakes.