How....

Let's use x^2 + 10x + 25 as our example for factorisation and expansion of quadratic expressions. 

First, using the cross-multiplication method . Draw two lines as shown below :

Then , write down the figures , in this case x^2 , 25 and 10x . Always remember that the variable with only one power should be at the right.

Choose factors to write down on top of the numbers like in this case : x x x = x^2 and 5 x 5 =25

Cross-multiply the factors according to the arrows shown below : (x x 5 = 5x)

On the right side write down the product which is in this case is 5x. Then make sure that the numbers at the right side add up to the number below. Meaning 5x + 5x = 10x , which verifies that this combination is correct. If the combination tested out is wrong , that means a wrong factor is chosen.

Based on this , we know that x^2 + 25 + 10x = (x+5)(x+5) which also means (x+5)^2

Thus , the factorised form is (x+5)^2 since there are brackets .

However , the multiplication frame cannot be used to expand. To expand , distributive law can be used . Distributive law is to multiply figures inside the brackets and those outside to open up the brackets. After using the distributive law , simplify and the correct expanded form should be obtained .

 Both factorisation and expansion and be done using special identities . There are 3 types of special identities : (a + b)^2 = a^2 +2ab + b^2 , (a-b)^2 = a^2 -2ab + b^2 and a^2 - b^2 = (a+b)(a-b)

Using the example earlier , x^2 + 10x + 25 would fit into the special identity : (a + b)^2 = a^2 +2ab + b^2 . In this case, a would be x while b would be 5 . Thus , to factorise using the special identity, (a+b)^2 would be (x+5)^2 , giving the correct factorised form. 

In converse , to expand, use the same special identities and change (x+5)^2 back to the expanded form which is x^2 + 10x + 25. However, special identities can only be used for quadratic expressions that can fit into one of these special identities which means if the equation does not fit , cross-multiplication method can be used to factorise and distributive law can be used to expand .

Expansion and Factorisation....

To change the form of algebraic expressions to make it simpler or more complicated, expansion and factorisation can be done.
A factorised form of algebraic expression should have brackets. Factorisation means pulling out the common factors and putting them into brackets. Example, 2x +6 = 2(x+3), 21y +3 =3(7y+1).
Expansion is the exact opposite of factorisation. While factorisation is putting in brackets, expansion is removing the brackets. To remove the brackets, the common factors are to be multiplied to the numbers in the brackets. Example, 2(x+3) = 2x +6, 3(7y +1) = 21y +3.
However, this method of factorisation and expansion is only applicable to algebraic expressions as quadratic equations hv more coefficients and powers of variables which makes it hard to factorise. Factorisation and expansion of quadratic equation can only be done using the cross method or special identities.

Quadratic Expressions.....

An expression has no equal sign. A basic algebraic expression is made up of algebraic terms together with numbers such as : x + 3 , z + 5. It is normally made up of a variable and a number. A standard algebraic equation would be by + c.
Whereas, a quadratic expression is made of up algebraic terms of two different powers together with numbers. Examples would be : x^2 + 7y +2 , 3x^2 + 2y + 6. A standard quadratic equation is : ax^2 + by + c.  
A quadratic equation is different from an algebraic expression as it has " ax^2 " that does not exist in algebraic expressions. Thus, in an quadratic expression, a cannot be equals to 0 or it will be an algebraic expression.