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Chapter 10: Problem 90

Factor. $$\left(4 x^{2}+12 x+9\right)-4 y^{2}$$

Short Answer

Expert verified
The factored form of the expression is \((2x+3+2y)(2x+3-2y)\).

Step by step solution

01

Identify the parts of the expression with square values

The expression given can be split into two parts: \((4x^2 + 12x + 9)\) and \(-4y^2\). The first part is a perfect square trinomial, because it can be written as \((2x+3)^2\), and the second part is the square of \(-2y\).
02

Factorising the Perfect Square Trinomial

The perfect square trinomial should be rewritten as a square. The trinomial \((4x^2 + 12x + 9)\) can be factored as \((2x+3)^2\). This factors out to \((2x+3)(2x+3)\).
03

Applying the difference of squares

The whole expression can now be seen as a difference of squares - the first square being \((2x+3)\) and the second square being \(2y\). This means we can factor it as \((2x+3+2y)(2x+3-2y)\), following the \((a+b)(a-b)\) pattern.

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Most popular questions from this chapter

For Exercises 144 to \(149,\) determine the positive integer values of \(k\) for which the polynomial is factorable over the integers. $$c^{2}-7 c+k$$

For Exercises 78 to \(131,\) factor completely. $$3 a^{3}-9 a^{2}-54 a$$

Factor by grouping. $$15 b^{2}-115 b+70$$

For Exercises 144 to \(149,\) determine the positive integer values of \(k\) for which the polynomial is factorable over the integers. $$z^{2}+7 z+k$$

For Exercises 78 to \(131,\) factor completely. $$4 a^{2}-40 a b+100 b^{2}$$
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