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# 3.Simplify: (i) $$(x^2 – 5) (x + 5) + 25$$ (ii) $$(a^2 + 5)(b^3 + 3) + 5$$ (iii) $$(t + s^2) (t^2 – s)$$ (iv)$$(a + b) (c – d) + (a – b) (c + d) + 2(ac + bd)$$ (v)$$(x + y) (2x + y) + (x + 2y) (x – y)$$ (vi) $$(x + y)(x^2 – xy + y^2)$$ (vii)$$(1.5x – 4y)(1.5x + 4y + 3) – 4.5x + 12y$$ (viii)$$(a + b + c) (a + b – c)$$

Answer :

(i)Simplifying the expression, we have: $$(x^2 – 5) (x + 5) + 25$$

$$= x^2 \times (x + 5) + 5\times (x + 5) + 25$$

$$= x^3 + 5x^2 – 5x – 25 + 25$$

$$= x^3 + 5x^2 – 5x + 0$$

$$= x^3 + 5x^2 – 5x$$

(ii)Simplifying the expression, we have: $$(a^2 + 5)(b^3 + 3) + 5$$

$$= a^2\times (b^3 + 3) + 5\times (b^3 + 3) + 5$$

$$= a^2b^3 + 3a^2 + 5b^3 + 15 + 5$$

$$= a^2b^3 + 3a^2 + 5b^3 + 20$$

(iii)Simplifying the expression, we have:$$(t + s^2) (t^2 – s)$$

$$= t\times (t^2 – s) + s^2 \times (t^2 – s)$$

$$= t^3 – st + s^2t^2 – s^3$$

$$= t^3 + s^2t^2 – st – s^3$$

(iv) Simplifying the expression, we have:$$(a + b)(c – d) + (a – b) (c + d) + 2(ac + bd)$$

$$= a\times(c – d) + b\times(c – d) + a\times(c + d) – b\times(c + d) + 2ac + 2bd$$

$$= ac – ad + bc – bd + ac + ad – bc – bd + 2ac + 2bd$$

$$= ac + ac + 2ac + bc – bc – ad + ad – bd – bd + 2bd$$

$$= 4ac + 0 + 0 + 0$$

$$= 4ac$$

(v) Simplifying the expression, we have:$$(x + y) (2x + y) + (x + 2y) (x – y)$$

$$= x\times(2x + y) + y\times(2x + y) + x\times(x – y) + 2y\times(x – y)$$

$$= 2x^2 + xy + 2xy + y^2 + x^2 – xy + 2xy – 2y^2$$

$$= 2x^2 + x^2 + xy + 2xy – xy + 2xy + y^2 – 2y^2$$

$$= 3x^2 + 4xy – y^2$$

(vi)Simplifying the expression, we have:$$(x + y)(x^2 – xy + y^2)$$

$$= x\times(x^2 – xy + y^2) + y\times (x^2 – xy + y^2)$$

$$= x^3 – x^2y + x^2y + xy^2 – xy^2 + y^3$$

$$= x^3 – 0 + 0 + y^3$$

$$= x^3 + y^3$$

(vii)Simplifying the expression, we have:$$(1.5x – 4y)(1.5x + 4y + 3) – 4.5x.+ 12y$$

$$= 1.5x \times (1.5x + 4y + 3) – 4y\times(1.5x + 4y + 3) – 4.5x + 12y$$

$$= 2.25x^2 + 6xy + 4.5x – 6xy – 16y^2 – 12y – 4.5x + 12y$$

$$= 2.25x^2 + 6xy – 6xy + 4.5x – 4.5x + 12y – 12y – 16y^2$$

$$= 2.25x^2 + 0 + 0 + 0 – 16y62$$

$$= 2.25x^2 – 16y^2$$

(viii)Simplifying the expression, we have: $$(a + b + c) (a + b – c)$$

$$= a(a + b – c) + b(a + b – c) + c(a + b – c)$$

$$= a^2 + ab – ac + ab + b^2 – bc + ac + bc – c^2$$

$$= a^2 + ab + ab – bc + bc – ac + ac + b^2 – c^2$$

$$= a^2 + 2ab + b^2 – c^2 + 0 + 0$$

$$= a^2 + 2ab + b^2 – c^2$$