I have a few math questions I made them just like my actual problems so that I can go through the steps you have went through to answer my own problems...Again THESE ARE NOT THE REAL PROBLEMS JUST EXAMPLES LIKE MINE WITH DIFFERENT NUMBERS. I don't know how to complete these problems so I made up some to learn how to do my actual problems.

I am unsure how to make a radical sign with the keyboard so I am using a question mark in its place.

(2?5-?28)(?28+4?5)
(?3+2?5)^2
(3?7-6?2)(7?7+?2)
(4x?y+2)(4x?y-?2)
(6?2-5?5)(3?2+2?5)
Thank you for helping.

For √, type in

& r a d i c ;
but without the intervening spaces.

I will solve the first one:
Use the FOIL rule
(2√5-√28)(√28+4√5)
=(2√5-2√7)(2√7+4√5)
=4(√5-√7)(√7+2√5)
=4(10-7-√5√7)
=12-4√5√7

To solve these problems, we'll need to know how to simplify expressions involving radicals. Let's go through each problem step by step and explain the process.

1) (2√5 - √28)(-√28 + 4√5):
To simplify this expression, we can use the distributive property, just like with regular algebra. We multiply the terms inside the first parentheses by both terms inside the second parentheses, and then combine like terms.

First, let's simplify the radicals within the parentheses:
√28 can be simplified as 2√7 since 28 can be factored as 2 * 2 * 7.

Now, we can rewrite the expression:
(2√5 - 2√7)(-2√7 + 4√5)

Next, we'll use the distributive property to multiply the terms within the parentheses:
= (2√5 * -2√7) + (2√5 * 4√5) + (-2√7 * -2√7) + (-2√7 * 4√5)

Simplifying each term:
= -4√35 + 8√25 + 4√49 - 8√35

Simplifying further:
= 8√25 - 4√35 - 8√35 + 4√49

= 8(5) - 4√35 - 8√35 + 4(7)

= 40 - 4√35 - 8√35 + 28

= 68 - 12√35

So, the simplified expression is 68 - 12√35.

2) (√3 + 2√5)^2:
To simplify this expression, we'll use the exponent property: (a + b)^2 = a^2 + 2ab + b^2.

Applying the formula:
(√3 + 2√5)^2 = (√3)^2 + 2(√3)(2√5) + (2√5)^2

Simplifying each term:
= 3 + 2(2)(√3)(√5) + 4(5)

= 3 + 4√15 + 20

= 23 + 4√15

So, the simplified expression is 23 + 4√15.

3) (3√7 - 6√2)(7√7 + √2):
Again, we'll use the distributive property to multiply the terms within the parentheses.

Simplifying the radicals within the parentheses:
√2 cannot be simplified any further.

Rewriting the expression:
(3√7 - 6√2)(7√7 + √2)

Multiplying the terms within the parentheses:
= (3√7 * 7√7) + (3√7 * √2) + (-6√2 * 7√7) + (-6√2 * √2)

Simplifying each term:
= 21√49 + 3√14 - 42√14 - 6(2)

= 21(7) + 3√14 - 42√14 - 12

= 147 - 39√14

So, the simplified expression is 147 - 39√14.

4) (4x√y + 2)(4x√y - √2):
Again, we'll use the distributive property to multiply the terms within the parentheses.

Simplifying the radicals within the parentheses:
√2 cannot be simplified any further.

Rewriting the expression:
(4x√y + 2)(4x√y - √2)

Multiplying the terms within the parentheses:
= (4x√y * 4x√y) + (4x√y * -√2) + (2 * 4x√y) + (2 * -√2)

Simplifying each term:
= 16x^2y - 4x√2 + 8x√y - 2√2

Combining like terms:
= 16x^2y + 8x√y - 4x√2 - 2√2

So, the simplified expression is 16x^2y + 8x√y - 4x√2 - 2√2.

5) (6√2 - 5√5)(3√2 + 2√5):
Using the distributive property again to multiply the terms within the parentheses.

Simplifying the radicals within the parentheses:
√2 and √5 cannot be simplified any further.

Rewriting the expression:
(6√2 - 5√5)(3√2 + 2√5)

Multiplying the terms within the parentheses:
= (6√2 * 3√2) + (6√2 * 2√5) + (-5√5 * 3√2) + (-5√5 * 2√5)

Simplifying each term:
= 18√4 + 12√10 - 15√10 - 10√25

Simplifying further:
= 18(2) + 12√10 - 15√10 - 10(5)

= 36 + 12√10 - 15√10 - 50

= -14 - 3√10

So, the simplified expression is -14 - 3√10.

I hope this step-by-step explanation helps you understand how to solve these types of problems!