Question

Factorise \[5\left( 3x + y \right)^{2} + 6\left( 3x + y \right) – 8\] .

Answer 1

Hint: In this question, we need to factorise the given expression \[5\left( 3x + y \right)^{2} + 6\left( 3x + y \right) – 8\] . To solve this question, first we can consider the term \[(3x+y)\] by a variable. Then we need to split the middle term of the expression. In order to split the middle term of the expression , we use the sum product form . We can factorise the given expression by taking the common terms outside. Then again we need to substitute \[(3x+y)\] in the expression. Using this we can easily factorise the given expression.
Sum product rule :
The general form of the polynomial is \[ax^{2} + bx + c = 0\]
The sum product form is the product of the middle term after splitting must be equal to \[a \times c\] and the sum must be equal to \[b\] .

Complete step-by-step answer:
Given, \[5\left( 3x + y \right)^{2} + 6\left( 3x + y \right) – 8\]
Let us consider the given expression as \[f(x)\]
\[f\left( x \right) = 5\left( 3x + y \right)^{2} + 6\left( 3x + y \right) – 8\]
Now we need to factorise the given expression by considering the term \[(3x + y)\] as \[a\] .
\[\Rightarrow \ 5a^{2} + 6a – 8\]
Now we can factorise the expression by splitting the middle term.
By sum product rule, the middle term 6a can split into \[10a\] and \[- 4a\] .
Now we can rewrite \[6a\] as \[(10a – 4a)\]
Thus the expression becomes
\[\Rightarrow \ 5a^{2} + 10a – 4a – 8\]
The expression consists of four terms. We can take \[5a\] common from the first two terms and also \[- 4\] common from the next two terms.
By taking the common terms outside,
We get,
\[\Rightarrow \ 5a(a + 2)\ 4(a + 2)\]
Again by taking \[(a + 2)\] common ,
We get,
\[\Rightarrow \ (5a – 4)\ (a + 2)\]
Now we can substitute the value of \[a\] ,
\[\Rightarrow \ (5(3x + y)\ - 4)\ (3x + y + 2)\] where \[a = 3x + y\]
On simplifying,
We get ,
\[f(x)\ = (15x + 5y – 4)\ (3x + y + 2)\]
Thus the factorisation of \[5\left( 3x + y \right)^{2} + 6\left( 3x + y \right) – 8\] is \[(15x + 5y – 4)\ (3x + y + 2)\] .
Final answer :
The factorisation of \[5\left( 3x + y \right)^{2} + 6\left( 3x + y \right) – 8\] is \[(15x + 5y – 4)\ (3x + y + 2)\] .

Note: An algebraic expression is nothing but it is a product of two simpler linear expressions. The concept used in this question to factorise the given expression by factorisation of a quadratic expression by Splitting the middle Term. Factorization is nothing but writing a whole number into smaller numbers of the same kind. In order to reduce the error , we should solve the question step by step wise, as it will help us to solve the question error-free . While opening the brackets , we need to make sure that we are opening the brackets properly with their respective signs.