a) x2 - 10x + 24
= x2 - 4x - 6x + 24
= x(x - 4) - 6(x - 4)
= (x - 4)(x - 6)
b) x2 - 13x + 36
= x2 - 4x - 9x + 36
= x(x - 4) - 9(x - 4)
= (x - 4)(x - 9)
a.
\(x^2-10x+24\)
\(=x^2-2\cdot x\cdot5+25-1\)
\(=\left(x-5\right)^2-1^2=\left(x-6\right)\left(x-4\right)\)
b.
\(x^2-13x+36\)
\(=x^2-2\cdot x\cdot\dfrac{13}{2}+\dfrac{169}{4}-\dfrac{25}{4}\)
\(=\left(x-\dfrac{13}{2}\right)^2-\left(\dfrac{5}{4}\right)^2=\left(x-\dfrac{21}{4}\right)\left(x-\dfrac{31}{4}\right)\)
c.
\(x^2-5x-24\)
\(=x^2-2\cdot x\cdot\dfrac{5}{2}+\dfrac{25}{4}-\dfrac{121}{4}\)
\(=\left(x-\dfrac{5}{2}\right)^2-\left(\dfrac{11}{2}\right)^2=\left(x-8\right)\left(x+3\right)\)
d.
\(x^3+3x^2-3x-1\)
\(=x^3-3x^2+3x-1+6x^2-6x\)
\(=\left(x-1\right)^3-6x\left(x-1\right)=\left(x-1\right)\left[\left(x-1\right)^2-6x\right]=\left(x-1\right)\left(x^2-8x+1\right)\)