To solve the problem of dividing the polynomial \( 5x^2 + 10x - 15 \) by \( x + 5 \), we can use polynomial division.
Here are the steps for polynomial division:
1. Set up the division:
- The polynomial to be divided is \( 5x^2 + 10x - 15 \) (the dividend).
- The divisor is \( x + 5 \).
2. Divide the first term of the dividend by the first term of the divisor:
- The first term of the dividend \( 5x^2 \) divided by the first term of the divisor \( x \) gives \( 5x \).
3. Multiply the entire divisor by the result:
- \( 5x \cdot (x + 5) = 5x^2 + 25x \).
4. Subtract the result from the original polynomial:
- Subtract \( 5x^2 + 25x \) from \( 5x^2 + 10x - 15 \):
[tex]\[
(5x^2 + 10x - 15) - (5x^2 + 25x) = (10x - 25x - 15) = -15x - 15
\][/tex]
5. Divide the first term of the new polynomial by the first term of the divisor \( x \):
- \(-15x \div x = -15\).
6. Multiply the entire divisor by \( -15 \):
- \( -15 \cdot (x + 5) = -15x - 75 \).
7. Subtract the result from the new polynomial:
- Subtract \(-15x - 75\) from \(-15x - 15\):
[tex]\[
(-15x - 15) - (-15x - 75) = -15 + 75 = 60
\][/tex]
So, the remainder is the constant term left after completing the polynomial division.
Thus, the remainder when the polynomial [tex]\( 5x^2 + 10x - 15 \)[/tex] is divided by [tex]\( x + 5 \)[/tex] is [tex]\( \boxed{60} \)[/tex].
