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To solve the given problem, you are asked to find the remainder when the polynomial \( 5x^2 + 10x - 15 \) is divided by \( x + 5 \).

This is a typical polynomial division problem, and we can use **the Remainder Theorem**. The Remainder Theorem states that the remainder of dividing a polynomial \( f(x) \) by \( x - a \) is equal to \( f(a) \).

In our case, the divisor is \( x + 5 \), which can be rewritten as \( x - (-5) \). According to the Remainder Theorem, we substitute \( x = -5 \) into the polynomial.

Let’s calculate \( f(-5) \) where \( f(x) = 5x^2 + 10x - 15 \):

\[
f(-5) = 5(-5)^2 + 10(-5) - 15
\]

Now, let's compute it step by step.

1. \( (-5)^2 = 25 \)
2. \( 5 \times 25 = 125 \)
3. \( 10 \times (-5) = -50 \)
4. \( 125 - 50 = 75 \)
5. \( 75 - 15 = 60 \)

So, the remainder when \( 5x^2 + 10x - 15 \) is divided by \( x + 5 \) is **60**.

Would you like more details or have any other questions?

Here are some related questions you might find interesting:
1. What is synthetic division and how can it help solve similar problems?
2. How does the Remainder Theorem work for higher degree polynomials?
3. Can the result of polynomial division ever be a fraction?
4. What is the relationship between the Factor Theorem and the Remainder Theorem?
5. How would the remainder change if the divisor were \( x - 3 \) instead?

**Tip:** The Remainder Theorem can save time in polynomial division problems by providing a quick way to find remainders without performing full division.

What is the remainder when the polynomial 5x^2 + 10x - 15 is divided by x + 5?

Find the remainder when the polynomial 5x^2 + 10x - 15 is divided by x + 5. Learn how to apply the Remainder Theorem for quick solutions in polynomial division. Suitable for students in Grades 9-10.

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