Simplifying (-4c - 7)(5c + 6): A Step-by-Step Guide

Hey guys! Let's dive into simplifying the expression (-4c - 7)(5c + 6). This kind of problem pops up a lot in algebra, and mastering it can really boost your math skills. We'll break it down step by step, so it’s super easy to follow. Think of it as unlocking a puzzle – each step gets us closer to the solution. So, grab your pencils, and let's get started!

Understanding the Basics: The Distributive Property

Before we jump into the main problem, let's quickly recap the distributive property. This is our key tool for simplifying expressions like this. The distributive property basically says that a(b + c) = ab + ac. In simpler terms, you multiply the term outside the parentheses by each term inside the parentheses. This principle is super important, and you'll use it all the time in algebra. It’s like the golden rule of simplifying expressions! So, make sure you have a solid grasp of it before moving on. Trust me, it will make everything else much easier. For instance, if we have 2(x + 3), we distribute the 2 to both x and 3, resulting in 2x + 6. This might seem simple, but it's the foundation for more complex simplifications.

Applying the Distributive Property to Our Problem

Now, let's see how the distributive property applies to our expression, (-4c - 7)(5c + 6). We essentially need to distribute each term in the first parenthesis to each term in the second parenthesis. This is also sometimes called the FOIL method (First, Outer, Inner, Last), which is just a handy way to remember the order of distribution. First, we multiply the first terms: (-4c) * (5c). Then, we multiply the outer terms: (-4c) * (6). Next, we multiply the inner terms: (-7) * (5c). And finally, we multiply the last terms: (-7) * (6). Writing it all out, we have: (-4c)(5c) + (-4c)(6) + (-7)(5c) + (-7)(6). See how we're just taking each term and pairing it up? This is the core of the distributive property in action, and it's what allows us to expand and simplify the expression. It’s like a meticulous dance, ensuring each term gets its turn in the spotlight.

Step-by-Step Breakdown of the Expansion

Let's break down each multiplication step by step to make sure we don't miss anything. First, we have (-4c)(5c). Multiplying the coefficients, -4 and 5, gives us -20. Then, multiplying c by c gives us c². So, (-4c)(5c) = -20c². Next up is (-4c)(6). Here, we multiply -4 by 6, which gives us -24. So, (-4c)(6) = -24c. Moving on to (-7)(5c), we multiply -7 by 5, which gives us -35. Thus, (-7)(5c) = -35c. Lastly, we have (-7)(6), which is simply -42. So, (-7)(6) = -42. Now, let's put it all together: Our expanded expression is -20c² - 24c - 35c - 42. We’ve successfully navigated the multiplication phase, and now we're ready to tidy things up. Think of it as decluttering – we’re making the expression cleaner and easier to handle.

Combining Like Terms: Simplifying Further

Now that we've expanded the expression, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expanded expression, -20c² - 24c - 35c - 42, the like terms are -24c and -35c, as they both have the variable c raised to the power of 1. The term -20c² has c raised to the power of 2, so it's not a like term with the others. And -42 is a constant term, so it's in a category of its own. To combine -24c and -35c, we simply add their coefficients: -24 + (-35) = -59. So, -24c - 35c = -59c. Remember, combining like terms is like grouping similar objects together – you can only combine what's the same. It’s a fundamental simplification technique, and mastering it will make complex algebraic manipulations much smoother.

Putting It All Together: The Final Simplified Expression

After combining like terms, our expression becomes -20c² - 59c - 42. This is the simplified form of the original expression, (-4c - 7)(5c + 6). We started with a product of two binomials and, through the distributive property and combining like terms, we've arrived at a trinomial. This final expression is much cleaner and easier to work with in further calculations. It’s like taking a messy room and organizing it – everything is now in its place, and it’s much easier to find what you need. So, our journey from (-4c - 7)(5c + 6) to -20c² - 59c - 42 is complete! We’ve successfully simplified the expression, and you’ve leveled up your algebra skills.

Common Mistakes to Avoid

Alright, let's chat about some common pitfalls people stumble into when simplifying expressions like this. Knowing these can save you from making errors yourself. One frequent mistake is messing up the signs. Remember, a negative times a negative is a positive, and a negative times a positive is a negative. Keep those rules locked in! Another common error is only distributing to the first term in the parentheses. Make sure you multiply each term inside the parentheses by the term outside. And, of course, don't forget to combine like terms at the end! It's easy to overlook this step, but it's crucial for getting the fully simplified answer. Also, be careful when multiplying variables; remember that c times c is c², not 2c. Keeping these common errors in mind can help you tackle these problems with confidence and accuracy. It’s like knowing the traps in a game – you can avoid them and win!

Practice Problems to Sharpen Your Skills

Okay, you've got the theory down, but the real magic happens with practice! Let's try a few more examples to solidify your understanding. Grab a piece of paper and a pencil, and let's work through them together. Here are a couple of expressions you can try simplifying:

1. (3x + 2)(2x - 1)
2. (-5y - 4)(y + 3)

Work through these using the steps we've discussed: distribute, multiply, and combine like terms. Don't worry if you don't get it right away; the key is to keep practicing. The more you do, the more comfortable you'll become with the process. It’s like learning a new dance – the more you rehearse, the smoother your moves become. So, give these problems a shot, and let's turn you into a simplification superstar!

Conclusion: Mastering Algebraic Simplification

So, there you have it! We've walked through how to simplify the expression (-4c - 7)(5c + 6), and you've learned some awesome skills along the way. Simplifying expressions like this is a fundamental part of algebra, and it's something you'll use again and again. By understanding the distributive property, combining like terms, and avoiding common mistakes, you're well on your way to mastering algebraic simplification. Remember, practice makes perfect, so keep working at it, and you'll become a pro in no time. It’s like building a house – each skill you learn is a brick that makes your foundation stronger. Keep building, and you’ll have a solid understanding of algebra in no time!