Factorize: \( \begin{array}{lll}\text { (a) } a^{2}+12 a+36 & \text { (b) } y^{2}+14 y+49 & \text { (c) } p^{2}+22 p+121 \\ \text { (d) } 4 a^{2}+20 a+25 & \text { (e) } 9 r^{2}+60 \mathrm{r}+100 & \text { (f) } 36 x^{2}+84 x+49 \\ \text { (g) } x^{2}-8 x+16 & \text { (h) } a^{2}-18 a+81 & \text { (i) } p^{2}-26 p+169\end{array} \)

To factorize the given expressions, we will look for perfect square trinomials and apply the appropriate factoring techniques. ### (a) \( a^{2}+12 a+36 \) This is a perfect square trinomial: \[ a^{2}+12 a+36 = (a+6)^{2} \] ### (b) \( y^{2}+14 y+49 \) This is also a perfect square trinomial: \[ y^{2}+14 y+49 = (y+7)^{2} \] ### (c) \( p^{2}+22 p+121 \) This is another perfect square trinomial: \[ p^{2}+22 p+121 = (p+11)^{2} \] ### (d) \( 4 a^{2}+20 a+25 \) First, factor out the common factor of 4: \[ 4(a^{2}+5 a+\frac{25}{4}) = 4(a^{2}+5 a+\left(\frac{5}{2}\right)^{2}) = 4\left(a+\frac{5}{2}\right)^{2} \] ### (e) \( 9 r^{2}+60 r+100 \) First, factor out the common factor of 1: \[ 9(r^{2}+\frac{60}{9} r+\frac{100}{9}) = 9\left(r^{2}+\frac{20}{3} r+\left(\frac{10}{3}\right)^{2}\right) = 9\left(r+\frac{10}{3}\right)^{2} \] ### (f) \( 36 x^{2}+84 x+49 \) First, factor out the common factor of 1: \[ 36(x^{2}+\frac{84}{36} x+\frac{49}{36}) = 36\left(x^{2}+\frac{7}{3} x+\left(\frac{7}{6}\right)^{2}\right) = 36\left(x+\frac{7}{6}\right)^{2} \] ### (g) \( x^{2}-8 x+16 \) This is a perfect square trinomial: \[ x^{2}-8 x+16 = (x-4)^{2} \] ### (h) \( a^{2}-18 a+81 \) This is a perfect square trinomial: \[ a^{2}-18 a+81 = (a-9)^{2} \] ### (i) \( p^{2}-26 p+169 \) This is a perfect square trinomial: \[ p^{2}-26 p+169 = (p-13)^{2} \] ### Summary of Factorizations: - (a) \( (a+6)^{2} \) - (b) \( (y+7)^{2} \) - (c) \( (p+11)^{2} \) - (d) \( 4\left(a+\frac{5}{2}\right)^{2} \) - (e) \( 9\left(r+\frac{10}{3}\right)^{2} \) - (f) \( 36\left(x+\frac{7}{6}\right)^{2} \) - (g) \( (x-4)^{2} \) - (h) \( (a-9)^{2} \) - (i) \( (p-13)^{2} \)