The problem you're working on asks to add the following expressions using the horizontal method by combining like terms. Let's go through each one.

a) $x^3 - x^2 + 7$

There are no like terms to combine here, so the expression remains the same.

b) $2x^2 - 4x - 3$ and $9x^2 - 9x + 10x^2 - 6$

Combining like terms:

• $2x^2 + 9x^2 + 10x^2 = 21x^2$
• $-4x - 9x = -13x$
• $-3 - 6 = -9$

So the result is $21x^2 - 13x - 9$.

c) $2x^2 + 6x + 15$ and $-15x^2 + 10x - 5$

Combining like terms:

• $2x^2 - 15x^2 = -13x^2$
• $6x + 10x = 16x$
• $15 - 5 = 10$

So the result is $-13x^2 + 16x + 10$.

d) $10x^2 - 15x + 12$ and $14x^2 - 5x + 16$

Combining like terms:

• $10x^2 + 14x^2 = 24x^2$
• $-15x - 5x = -20x$
• $12 + 16 = 28$

So the result is $24x^2 - 20x + 28$.

If you'd like more explanation or if you have other questions about this, feel free to ask!

Here are some related questions:

1. How do you combine like terms when dealing with more than two expressions?
2. What is the significance of the degree of a term in an expression when combining terms?
3. How would you approach adding polynomials that have different variables?
4. Can you explain how to subtract polynomials using the same horizontal method?
5. What is the result when you add two polynomials that have no common terms?

Tip: When combining terms, make sure the terms you add have the same variable and degree; otherwise, they can't be combined.