Begin by distributing each part:

2x(x² + 4ax – 3a²) = 2x * x² + 2x * 4ax – 2x * 3a² = 2x³ + 8ax² – 6a²x

The second:

–4a²(4x + 3a) = –16a²x – 12a³

Now, combine these:

2x³ + 8ax² – 6a²x – 16a²x – 12a³

The only common terms are those with a²x; therefore, this reduces to

2x³ + 8ax² – 22a²x – 12a³

This is the same as the given answer:

–12a³ – 22a²x + 8ax² + 2x³

To simplify this, we will want to use the correct order of operations. The mnemonic device PEMDAS is usually very helpful.

Parenthesis (1st)

Exponents (2nd)

Multiply, Divide (3rd)

Add, Subtract (4th)

PEMDAS tells us to evaluate parentheses first, then exponents. After exponents, we evaluate multiplication and division from left to right, and then we evaluate addition and subtraction from left to right.

Let's look at (xy)² – x((4x)(y)² – (4x)²) – 4²x²

We want to start with parentheses first. We will simplify the (xy)² by using the general rule of exponents, which states that (ab)^c = a^cb^c. Thus we can replace (xy)² with x²y².

x²y² – x((4x)(y)² – (4x)²) – 4²x²

When we have parentheses within parentheses, we want to move from the innermost parentheses to the outermost. This means we will want to simplify the expression (4x)(y)² – (4x)² first, which becomes 4xy² – 16x². We can now replace (4x)(y)² – (4x)² with 4xy² – 16x² .

x²y² – x(4xy² – 16x²) – 4²x²

In order to remove the last set of parentheses, we will need to distribute the x to  4xy² – 16x² . We will also make use of the property of exponents which states that a^ba^c = a^b+c.

x²y² – x(4xy²) – x(16x² ) – 4²x²
= x²y² – 4x²y² – 16x³ – 4²x²

We now have the parentheses out of the way. We must now move on to the exponents. Really, the only exponent we need to simplify is –4², which is equal to –16. Remember that –4² = –(4²), which is not the same as (–4)².

x²y² – 4x²y² – 16x³ – 16x²

Now, we want to use addition and subtraction. We need to add or subtract any like terms. The only like terms we have are x²y² and –4x²y². When we combine those, we get –3x²y²

–3x²y² – 16x³ – 16x²

The answer is –3x²y² – 16x³ – 16x² .

 can also be expressed as 

5x – (6x – 2x) = 5x – (4x) = x

(x – 1)(x + 2) - x² + 2 = x² + x – 2 – x² + 2 = x

x(4x)/(4x) = x

(3 – 3)x = 0x = 0

Step 1: 7x + 3y

Step 2: 4 * (7x + 3y) = 28x + 12y

Step 3: 28x + 12y + x = 29x + 12y

Step 4: 29x + 12y – (x – y) = 29x + 12y – x + y = 28x + 13y

Use PEMDAS to dictate which operation comes first. Simplify the parentheses:
  and

.

Next come exponents:

After that comes multiplication and division left to right:

 and

.

Finally, add all the terms together:

To simplify this problem, let’s look at each term individually. ; ; . Thus B is the correct answer.

The problem gives us the product of two consecutive odd negative integers, so we know that one number is  less than the other one. Thus, we can set our two numbers as  and .

At this point, the more algebraically inclined student might recognize that if , then the equation can be remade to say , and use the quadratic formula to solve.

But this is the ACT, and the faster method by far is to simply recognize that if the product of our two integers is , then  must be evenly divisible by our two integers. The only two choices we have that divide evenly into  are  and , making  the smaller number and our answer.

Make a simple algebra equation and test it against each combination:

Total Cost = $2.50 * (# Oranges) +  $1.50 * (# Apples) +  $0.50 * (# Bananas)