QuestionFactorise fully $4 a x^{2} + 12 b x^{2} - 3 b - a$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to factorize the expression $4 a x^{}-3 b+12 b x^{}-a$ . . The expression is a polynomial.
3. The variables $a$ , $b$ , and $x$ are real numbers.

STEP 2

First, we need to rearrange the terms in the expression. It's easier to factorize when like terms are together.
$4 a x^{2}- b+12 b x^{2}-a =4 a x^{2} - a +12 b x^{2} - b$

STEP 3

Now, we can factor out the common factors from each pair of terms.
$a x^{2} - a +12 b x^{2} -3 b = a(x^{2} -1) + b(12x^{2} -3)$

STEP 4

Now, we can simplify the expression inside the brackets.
$a(4x^{2} -1) + b(12x^{2} -3) = a(2x -1)(2x +1) +3b(4x^{2} -1)$

STEP 5

We can see that the terms $(2x -1)(2x +1)$ and $(4x^{2} -1)$ are the same. We can factor this term out.
$a(2x -1)(2x +1) +3b(4x^{2} -1) = (2x -1)(2x +1)(a +3b)$ So, the fully factorized form of $4 a x^{2}-3 b+12 b x^{2}-a$ is $(2x -1)(2x +1)(a +3b)$ .

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