Factorise: \[{x^2} + xy + 8x + 8y\].

Hint:
Here, we need to factorise the given expression. First, we will group the terms in the given expression using parentheses. Then, we will factor out terms from the parentheses to rewrite the given algebraic expression as a product of two or more expressions, and hence, factorise it.

Complete step by step solution:
We have to factorise the expression \[{x^2} + xy + 8x + 8y\].
We will factor out common terms from the expression to factorise it.
Grouping the terms in the expression using parentheses, we get
\[ \Rightarrow {x^2} + xy + 8x + 8y = \left( {{x^2} + xy} \right) + \left( {8x + 8y} \right)\]
We can observe that \[{x^2}\] and \[xy\] have the common factor \[x\].
Therefore, we can factor out \[x\] from the terms \[{x^2}\] and \[xy\].
Factoring out \[x\] from the terms in the first parentheses, we get
\[ \Rightarrow {x^2} + xy + 8x + 8y = x\left( {x + y} \right) + \left( {8x + 8y} \right)\]
We can observe that \[8x\] and \[8y\] have the common factor 8.
Therefore, we can factor out 8 from the terms \[8x\] and \[8y\].
Factoring out 8 from the terms in the second parentheses, we get
\[ \Rightarrow {x^2} + xy + 8x + 8y = x\left( {x + y} \right) + 8\left( {x + y} \right)\]
Now, we can observe that \[x\left( {x + y} \right)\] and \[8\left( {x + y} \right)\] have the common factor \[\left( {x + y} \right)\].
Therefore, we can factor out \[\left( {x + y} \right)\] form the terms \[x\left( {x + y} \right)\] and \[8\left( {x + y} \right)\].
Factoring out \[\left( {x + y} \right)\] from the terms in the expression, we get
\[ \Rightarrow {x^2} + xy + 8x + 8y = \left( {x + y} \right)\left( {x + 8} \right)\]
This is the factored form of the given expression.

Therefore, we have factored the given expression \[{x^2} + xy + 8x + 8y\] as \[\left( {x + y} \right)\left( {x + 8} \right)\].

Note:
We factorised the given algebraic expression in the solution. Factorisation is the process of writing an equation as a product of its factors. We factored the algebraic expression \[{x^2} + xy + 8x + 8y\] as \[\left( {x + y} \right)\left( {x + 8} \right)\]. This means that \[\left( {x + y} \right)\] and \[\left( {x + 8} \right)\] are the factors of the algebraic expression \[{x^2} + xy + 8x + 8y\].
We can group the terms differently to obtain the same answer.
Grouping the terms in the expression using parentheses, we get
\[ \Rightarrow {x^2} + xy + 8x + 8y = \left( {{x^2} + 8x} \right) + \left( {xy + 8y} \right)\]
Factoring out \[x\] from the terms in the first parentheses, we get
\[ \Rightarrow {x^2} + xy + 8x + 8y = x\left( {x + 8} \right) + \left( {xy + 8y} \right)\]
Factoring out \[y\] from the terms in the second parentheses, we get
\[ \Rightarrow {x^2} + xy + 8x + 8y = x\left( {x + 8} \right) + y\left( {x + 8} \right)\]
Factoring out \[\left( {x + 8} \right)\] from the terms in the expression, we get
\[ \Rightarrow {x^2} + xy + 8x + 8y = \left( {x + 8} \right)\left( {x + y} \right)\]
Therefore, we have factored the given expression \[{x^2} + xy + 8x + 8y\] as \[\left( {x + 8} \right)\left( {x + y} \right)\].