Question

1. (4a^{2}-\frac{9}{25}b^{2}) 5) (x^{2}+4x - 5)

Explanation:

1. Factor \(4a^{2}-\frac{9}{25}b^{2}\):

• Step 1: Recognize the difference - of - squares formula
• The difference - of - squares formula is \(x^{2}-y^{2}=(x + y)(x - y)\).
• Rewrite \(4a^{2}-\frac{9}{25}b^{2}\) as \((2a)^{2}-(\frac{3}{5}b)^{2}\), where \(x = 2a\) and \(y=\frac{3}{5}b\).
• Then, by the difference - of - squares formula, \((2a)^{2}-(\frac{3}{5}b)^{2}=(2a+\frac{3}{5}b)(2a - \frac{3}{5}b)\).

1. Factor \(x^{2}+4x - 5\):

• Step 1: Find two numbers that multiply to \(ac=-5\) and add up to \(b = 4\)
• For the quadratic expression \(ax^{2}+bx + c=x^{2}+4x - 5\) (where \(a = 1\), \(b = 4\), \(c=-5\)), we need to find two numbers \(m\) and \(n\) such that \(m\times n=ac=-5\) and \(m + n=b = 4\).
• The numbers are \(m = 5\) and \(n=-1\) since \(5\times(-1)=-5\) and \(5+( - 1)=4\).
• Step 2: Rewrite the middle term and factor by grouping
• \(x^{2}+4x - 5=x^{2}+5x-x - 5\).
• Group the terms: \((x^{2}+5x)-(x + 5)\).
• Factor out the greatest common factor from each group: \(x(x + 5)-1(x + 5)\).
• Then, \((x + 5)(x - 1)\).

Answer:

1. \(4a^{2}-\frac{9}{25}b^{2}=(2a+\frac{3}{5}b)(2a - \frac{3}{5}b)\)
2. \(x^{2}+4x - 5=(x + 5)(x - 1)\)