Answer
a)
[tex]{(2b - 5)}^{2} - 36 =( 2b - 11)(2b + 1)[/tex]
b)
[tex]9 - {(7 + 3a)}^{2} = (3a - 4)(3a + 11)[/tex]
c)
[tex]( {4 - 11m)}^{2} - 1 =( 3 - 11m )( 5 - 11m )[/tex]
Explanation
a) The given expresion is
[tex] {(2b - 5)}^{2} - 36[/tex]
We rewrite as difference of two squares
[tex]{(2b - 5)}^{2} - 36 = {(2b - 5)}^{2} - {6}^{2} [/tex]
Recall that:
[tex] {x}^{2} - {y}^{2} = (x + y)(x - y)[/tex]
This implies that:
[tex]{(2b - 5)}^{2} - 36 =( {(2b - 5)} -6)(2b - 5 )+ 6)[/tex]
Or
[tex]{(2b - 5)}^{2} - 36 =( 2b - 5-6)(2b - 5 + 6)[/tex]
This simplifies to give:
[tex]{(2b - 5)}^{2} - 36 =( 2b - 11)(2b + 1)[/tex]
b) The second expression is
[tex]9 - {(7 + 3a)}^{2} [/tex]
We rewrite as perfect squares yo get:
[tex]9 - {(7 + 3a)}^{2} = {3}^{2} - {(7 + 3a)}^{2} [/tex]
This gives:
[tex]9 - {(7 + 3a)}^{2} = ({3} - {(7 + 3a)})({3} + {(7 + 3a)})[/tex]
This implies that
[tex]9 - {(7 + 3a)}^{2} = ({3} - 7 + 3a)({3} + 7 + 3a)[/tex]
We simplify to get:
[tex]9 - {(7 + 3a)}^{2} = (3a - 4)(3a + 11)[/tex]
c) The third expression is:
[tex]( {4 - 11m)}^{2} - 1[/tex]
We obtain the difference of two squares as:
[tex]( {4 - 11m)}^{2} - 1 =( ( {4 - 11m)} - 1 )( ( {4 - 11m)} + 1 )[/tex]
We simplify within the parenthesis to get:
[tex]( {4 - 11m)}^{2} - 1 =( 4 - 11m - 1 )( 4 - 11m+ 1 )[/tex]
We simplify further to get;
[tex]( {4 - 11m)}^{2} - 1 =( 3 - 11m )( 5 - 11m )[/tex]
