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Answer
The expression factors to \( (2x + 9y)(2x + y) \).
Solution
Factor the expression by following steps:
- step0: Factor:
\(4x^{2}+20xy+9y^{2}\)
- step1: Rewrite the expression:
\(4x^{2}+\left(2+18\right)xy+9y^{2}\)
- step2: Calculate:
\(4x^{2}+2xy+18yx+9y^{2}\)
- step3: Rewrite the expression:
\(2x\times 2x+2xy+9y\times 2x+9y\times y\)
- step4: Factor the expression:
\(2x\left(2x+y\right)+9y\left(2x+y\right)\)
- step5: Factor the expression:
\(\left(2x+9y\right)\left(2x+y\right)\)
The expression \( 4 x^{2}+20 x y+9 y^{2} \) can be factored as:
\[
(2x + 9y)(2x + y)
\]
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Simplify this solution Extra Insights
To factor the expression \( 4 x^{2}+20 x y+9 y^{2} \), we start by looking for two numbers that multiply to \( 4 \times 9 = 36 \) (the product of the coefficients of \( x^2 \) and \( y^2 \)) and add up to \( 20 \) (the coefficient of \( xy \)). The numbers \( 18 \) and \( 2 \) meet this requirement. Now we can rewrite the middle term \( 20xy \) using \( 18xy + 2xy \): \[ 4x^2 + 18xy + 2xy + 9y^2. \] Next, we can group the terms: \[ (4x^2 + 18xy) + (2xy + 9y^2). \] Factoring each group gives us: \[ 2x(2x + 9y) + y(2x + 9y). \] Now, we can factor out the common term \( (2x + 9y) \): \[ (2x + 9y)(2x + y). \] Thus, the factored expression is: \[ (2x + 9y)(2x + y). \]
