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factor each of the following expressions.
a ( 25 x^{2}+40 x + 16=)
b ( 36 x^{2}-84 x + 49=)
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Explanation:

Step1: Recall the perfect - square formula

The perfect - square formula is \((a + b)^2=a^{2}+2ab + b^{2}\) and \((a - b)^2=a^{2}-2ab + b^{2}\)

Step2: Factor \(25x^{2}+40x + 16\)

For the expression \(25x^{2}+40x + 16\), we have \(a^{2}=25x^{2}\), so \(a = 5x\); \(b^{2}=16\), so \(b = 4\). And \(2ab=2\times5x\times4 = 40x\).
By the perfect - square formula \((a + b)^2=a^{2}+2ab + b^{2}\), \(25x^{2}+40x + 16=(5x + 4)^2\)

Step3: Factor \(36x^{2}-84x + 49\)

For the expression \(36x^{2}-84x + 49\), we have \(a^{2}=36x^{2}\), so \(a = 6x\); \(b^{2}=49\), so \(b = 7\). And \(2ab=2\times6x\times7=84x\)
By the perfect - square formula \((a - b)^2=a^{2}-2ab + b^{2}\), \(36x^{2}-84x + 49=(6x - 7)^2\)

Answer:

a. \((5x + 4)^2\)
b. \((6x - 7)^2\)