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The distributive property is a fundamental concept in algebra that helps in simplifying expressions. It allows you to multiply a single term by each term within a set of parentheses. This means that if you have an expression like \( a(b + c) \), you can "distribute" \( a \) to both \( b \) and \( c \), resulting in the expression \( ab + ac \).

In the problem \( 6(-3) + 5(-9) \), the distributive property isn't directly applied like \( 6(x + y) \), but understanding how multiplication distributes over addition (even if implicit) helps solve the problem. Both \( 6 \) distributes over \( (-3) \) and \( 5 \) over \((-9)\) separately before performing addition, making it simpler to tackle such expressions.

Multiplication of integers might seem straightforward, but special care is needed particularly when negative numbers are involved. The general rules for multiplying integers are:

• If both numbers are positive, multiply them as usual.
• If one number is positive and the other is negative, the result is negative.
• If both numbers are negative, the result is positive.

In our exercise, \( 6(-3) \) and \( 5(-9) \) are examples where a positive integer is multiplied by a negative integer, resulting in negative products:

• \( 6 \times (-3) = -18 \).
• \( 5 \times (-9) = -45 \).

By knowing these rules, you can quickly find the product of any two integers, ensuring accuracy in simplification tasks.

When adding integers, particularly negatives, it is helpful to think about the number line. Adding a negative number is like moving left on the number line, while adding a positive number means moving right.

The specific rules are simple:

• Adding two positive integers results in a positive sum.
• Adding two negative integers gives a more negative sum.
• Adding a positive and a negative integer involves subtracting the smaller absolute value from the larger one and taking the sign of the number with the larger absolute value.

In this problem, we have:

• \(-18 + (-45) \)

Simply add the absolute values \(18 + 45\), which equals \(63\), and keep the negative sign, providing the answer \(-63\).

By understanding how negative numbers interact with others, you can accurately add integers in various math problems.